Measure theory

Table of Contents

Notation

  • measure_theory_8cb94a32b93fca2e76396dd9b96c4be2760eacba.png and measure_theory_c610f583621085e7f72317810e0dd15325091f66.png are used to denote the indicator or characteristic function

Definition

Motivation

The motivation behind defining such a thing is related to the Banach-Tarski paradox, which says that it is possible to decompose the 3-dimensional solid unit ball into finitely many pieces and, using only rotations and translations, reassemble the pieces into two solid balls each with the same volume as the original. The pieces in the decomposition, constructed using the axiom of choice, are non-measurable sets.

Informally, the axiom of choice, says that given a collecions of bins, each containing at least one object, it's possible to make a selection of exactly one object from each bin.

Measure space

If measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is a set with the sigma-algebra measure_theory_017948b866be67b1a8e56a6b5f8848f823410c34.png and the measure measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png, then we have a measure space .

Product measure

Given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space.

A product measure measure_theory_201435ab560b9faa34976b4b74b3c3e80c29a5ae.png is defined to be a measure on the measurable space measure_theory_4abd990c6ed8a4f055141a7f7675305ce09d585a.png, where we've let measure_theory_60b59bddc7486649ac40ab7d4fbb8bf58c4db3eb.png be the algebra on the Cartesian product measure_theory_426801f6f272b4f2bda3fd2f4bb41a4937108eb0.png. This sigma-algebra is called the tensor-product sigma-algebra on the product space.

A product measure measure_theory_201435ab560b9faa34976b4b74b3c3e80c29a5ae.png is defined to be a measure on the measurable space measure_theory_4abd990c6ed8a4f055141a7f7675305ce09d585a.png satisfying the property

measure_theory_8d0228c1c42ef424b8d3ddb8720621a9d3f8674e.png

measure_theory_e92373cc4b63bb94d271e994a969a155d96bdc02.png and measure_theory_37a0782c7c277f7e3d25d466e2494d705792f040.png

Let measure_theory_09199746b987edcf41d208cb6ef552607d577d81.png be a sequence of extended real numbers.

The limit inferior is defined

measure_theory_57c60c2f2d1c16e854be51c8d4accb9d18e8c9dc.png

The limit supremum is defined

measure_theory_c879430317d102ee993fd2e36bdcfda81c5117fa.png

Premeasure

Given a space measure_theory_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png, and a collection of sets measure_theory_fcd06e275c4c079a752ad55ec0d73c03ff66f7b1.png is an algebra of sets on measure_theory_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png if

  • measure_theory_76f45ab40d94f33ded4386a99a7270232cbed4e1.png
  • If measure_theory_41e6ce2ab80715fe484d74efc80263e1528225f7.png, then measure_theory_1d08ce3de72b91b62a5655ce70ebb8e258b4e0f0.png
  • If measure_theory_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png and measure_theory_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png are in measure_theory_37d3d2a40172b2cf7b0e8ea91ec033c3e1f9ecbd.png, then measure_theory_e0b531a2e964293565e81900dc2be4b53dd80ddf.png

Thus, a algebra of sets allow only finite unions, unlike σ-algebras where we allow countable unions.

Given a space measure_theory_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png and an algebra measure_theory_37d3d2a40172b2cf7b0e8ea91ec033c3e1f9ecbd.png, a premeasure is a function measure_theory_1498d7b81a50c283ec2af1f6805d25ad27408faf.png such that

  • measure_theory_594d2b656399f6c69afcdfcfe81d05b304aa471d.png
  • For every finite or countable collection of disjoint sets measure_theory_219082048e55b3ba3c5498379b1c3f570cda13ab.png with measure_theory_602479be5e5c6c6898e0b72acd59e63e0f6f6213.png, if measure_theory_c79647d752cb4da3bf46a1d49866583990b4c3ff.png then

    measure_theory_422d2bf861c82942fc27e883dc1fbc34129f6699.png

Observe that the last property says that IF this "possibly large" union is in the algebra, THEN that sum exists.

A premeasure space is a triple measure_theory_b88799d5ec72ed146276eedc13cdcd9204986c5c.png where measure_theory_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png is a space, measure_theory_37d3d2a40172b2cf7b0e8ea91ec033c3e1f9ecbd.png is an algebra, and a premeasure measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png.

Complete measure

A complete measure (or, more precisely, a complete measure space ) is a measure space in which every subset of every null set is measurable (having measure zero).

More formally, measure_theory_b55e28fa3e70a49dad8c382a0a81c02bba5551cf.png is complete if and only if

measure_theory_7bfd73be4edf8cb60b0f54ea329d654e1dcdc1cb.png

If measure_theory_c782ba1fbef42c8490f3b7cdc3c7ee4fa76e02a2.png is a premeasure space, then there is a complete measure space measure_theory_6966d1f429c6af0040c35ae3fdb7abfe36777624.png such that

  • measure_theory_7f9173178b700f632c2583528b3d25d5cc19cf30.png
  • measure_theory_c00b2433dd64e3b2b17f3b9dc88ea36d1bc9ee48.png we have measure_theory_afe94670cf32f8a11b1c3553dfff145681da0b2d.png

If measure_theory_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png is σ-finite, then measure_theory_fb8a0eeb838b8cf5de1075f3e0ab0ed94de9ffed.png is the only measure on measure_theory_e3dd79e5841f6fb0cf9667c279557d8489ed579d.png that is equal to measure_theory_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png on measure_theory_37d3d2a40172b2cf7b0e8ea91ec033c3e1f9ecbd.png.

Sobolev space

Notation

  • measure_theory_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png is an open subset of measure_theory_004097ff73cb85a0f596c8a3b60218ece0e16be1.png
  • measure_theory_639a884166da121d75367faf8e90ba8f1fcbeea7.png denotes a infinitively differentiable function measure_theory_078b85cd3478400338e3a1ee425c2a468644be7e.png with compact support
  • measure_theory_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png is a multi-index of order measure_theory_3a4189a25e6c0ad201e3ae2a133fdd9e6fffdab0.png, i.e.

    measure_theory_133c1389abe887c4224b062ad4751431be22d788.png

Definition

Vector space of functions equipped with a norm that is a combination of measure_theory_5bda34178cc82d1588df2a197af81c5f47d96d87.png norms of the function itself and its derivatoves to a given order.

Intuitively, a Sobolev space is a space of functions with sufficiently many derivatives for some application domain, e.g. PDEs, and equipped with a norm that measures both size and regularity of a function.

The Sobolev space spaces measure_theory_e82554e3d1082fd9ce05ec7527b38e2f5029f377.png combine the concepts of weak differentiability and Lebesgue norms (i.e. measure_theory_5bda34178cc82d1588df2a197af81c5f47d96d87.png spaces).

For a proper definition for different cases of dimension of the space measure_theory_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png, have a look at Wikipedia.

Motivation

Integration by parst yields that for every measure_theory_38819d1d6c06b8aa737e986909ceae6dc478655b.png where measure_theory_4028f9016d4f7a233195bc931939fa3360d350cd.png, and for all infinitively differentiable functions with compact support measure_theory_639a884166da121d75367faf8e90ba8f1fcbeea7.png:

measure_theory_395ae961a7b4560cd48e4b1e1037e7f73b929f59.png

Observe that LHS only makes sense if we assume measure_theory_8b0e7a5ea4d56f5afdac3c2658ca87bd50b0ad6f.png to be locally integrable. If there exists a locally integrable function measure_theory_d198c0d06b0da5525f9a966222601066509ac01d.png, such that

measure_theory_fbf05489ec587bf169ce19dcf6c3d42f801937a7.png

we call measure_theory_d198c0d06b0da5525f9a966222601066509ac01d.png the weak measure_theory_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png -th partial derivative of measure_theory_8b0e7a5ea4d56f5afdac3c2658ca87bd50b0ad6f.png. If this exists, then it is uniquely defined almost everywhere, and thus it is uniquely determined as an element of a Lebesgue space (i.e. measure_theory_5bda34178cc82d1588df2a197af81c5f47d96d87.png function space).

On the other hand, if measure_theory_38819d1d6c06b8aa737e986909ceae6dc478655b.png, then the classical and the weak derivative coincide!

Thus, if measure_theory_f33b408b8aed4567e7198c9a0bb79e3bbde50e04.png, we may denote it by measure_theory_b7fdf1ec978c965b42e6e55ef35243184c58a848.png.

Example

measure_theory_93f302a90840357efcd526fcdcfb4371d1bba5fb.png

is not continuous at zero, and not differentiable at −1, 0, or 1. Yet the function

measure_theory_08951f8c0ba6006aa8725bcce741fd587043e16a.png

satisfies the definition of being the weak derivative of measure_theory_d318a5efb8e9c28dd4bb1fead2cc1b6c4684a13f.png, which then qualifies as being in the Sobolev space measure_theory_81acc4334be481255fc9139da6d9f89a50e19e3d.png (for any allowed measure_theory_7225b076f6e6326f1636b11d1aad8de58bcc4761.png).

Lebesgue measure

Notation

  • measure_theory_22edaba801d6706cf2d1dcb13f9b6aab563b359a.png denotes the collection of all measurable sets

Stuff

Given a subset measure_theory_f9bf6d664d9d2a949a6d9944de07b272a08dcb57.png, with the length of a closed interval measure_theory_8de9f615a64b2e89e1f5c33cd1153517015b6a73.png given by measure_theory_f0ae4401c60408cdcc680328c83e65725e291a1f.png, the Lebesgue outer measure measure_theory_9c5a838dfd45201e3f983614c4fbae1348940f16.png is defined as

measure_theory_43981a17509f3551c0edb6851f1fc7958ed186b3.png

Lebesgue outer-measure has the following properties:

  1. measure_theory_965905726c53baa096e30fb14d35ccb1717b176b.png Idea: Cover by measure_theory_cda84ced58c9669648ffb80607a79c1264bca81a.png.
  2. (Monotinicy) measure_theory_bbb88a427218aaf3e1d873274c74932f32cfbb6b.png if measure_theory_53d5199519d69d64aa75e6032a18a2cf16d125ba.png, then

    measure_theory_6d69cf29784dc0c8f84126919214c28391ba1ce4.png

    Idea: a cover of measure_theory_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png is a cover of measure_theory_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png.

  3. (Countable subadditivity) For every set measure_theory_4a5410e91421c8a51784142a706c63339c4c695e.png and every sequence of sets measure_theory_3737972b1891f563de37e8da957242a1add3a432.png if measure_theory_8dd001f87e1413d9b533314e6051f3a516795748.png then

    measure_theory_dfc927b5b1a4775c442c48997647c544f8b9ceaa.png

    Idea: construct a cover of each measure_theory_f8b648069010a70233704a414386e26e49992dcb.png, measure_theory_74f56f17b71ac00a44785434a3ffe2652dd6983b.png such that measure_theory_d0b074c0342c715b44fdb9bdc05743e1910f2d2d.png:

    • Every point in measure_theory_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is in one of the measure_theory_f8b648069010a70233704a414386e26e49992dcb.png
    • measure_theory_bbf84e41538ffc52a38cc9bc4929ca62dc733e81.png

Q: Is it possible for every measure_theory_ebdcfdbbd4a497c69547e23cc3342abbf2127bfe.png to find a cover measure_theory_e43dc1818e282f7715042972e65e37cc020c77a6.png such that measure_theory_3b80f8127cc07537685b84750aed19e76b73b584.png? A: No. Consider measure_theory_45daf3c13bdff98253e0199590771b1c1a70e34e.png. Given measure_theory_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png, consider measure_theory_9698f9f7a082ac3403efd2c1859b5040029539c8.png. This is a cover of measure_theory_45daf3c13bdff98253e0199590771b1c1a70e34e.png so measure_theory_184fee159f84b50edc360254735b6ddd5d9d9ead.png. If measure_theory_9007ad37d5bec05fcaf476ad9e38d89a39caa8b2.png is a cover by open intervals of measure_theory_45daf3c13bdff98253e0199590771b1c1a70e34e.png, then there is at least one measure_theory_d8797d487554fc78e460f5843167fcb01b53e76b.png such that measure_theory_498b729fb14d87991858e5874a5dc2ba53bba82f.png is a nonempty open interval, so it has a strictly positive lenght, and

measure_theory_8b6bd0b92642122d1d5ff49b998b02df05e932cf.png

If measure_theory_892e884364fc2b126dc85629ee75372f91a6b711.png, then

measure_theory_419bdef0a4be7bcd9da8743a42cd8bad164f8d68.png

Idea: measure_theory_86dbf83ceba3dd4994b6db8f659f2c07afa82c3c.png, so measure_theory_eca9e34c537bcdba358ce1cd9112275da42334f2.png. For reverse, cover measure_theory_29de2d3d8043a6f75542dc782bf6a4d2119405c0.png by intervals giving a sum within measure_theory_224ae917d74dc2133d4403064c971bf562d4db50.png. Then cover measure_theory_af887f66eec166253cb70d37c5855092151b800d.png and measure_theory_66730768cb5ed69a50a2e181acf80c954c931569.png by intervals of length measure_theory_250ef6b6fb2f36c93e7e14a592a6f4ce375b8d92.png. Put the 2 new sets at at the start of the sequence, to get a cover of measure_theory_b2fdf0617d736136c4564c3795fed8545c0717bf.png, and sum of the lengths is at most measure_theory_f58d1e57238bafa35888d4aac509e901de17daa3.png. Hence,

measure_theory_578b139f21969e46aaa07cf97e5fd8b49c59aa90.png

If measure_theory_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png is an open interval, then measure_theory_e44a9da6d4ba40942ab8db89c3e02d673a3b2ba0.png.

Idea: lower bound from measure_theory_d30f0de02eaa3a2250c573a5201f1bcc72f57f64.png. Only bounded nonempty intervals are interesting. Take the closure to get a compact set. Given a countable cover by open intervals, reduce to a finite subcover. Then arrange a finite collection of intervals in something like increasing order, possibly dropping unnecessary sets. Call these new intervals measure_theory_530c493a33a6849627cea4b7ca3b80f4777a9aa3.png and let measure_theory_7225b076f6e6326f1636b11d1aad8de58bcc4761.png be the number of such intervals, and such that

measure_theory_2a1c29788ef8bb9fa95cffa30fab78078ce192e1.png

i.e. left-most interval cover the starting-point, and right-most interval cover the end-point. Then

measure_theory_50becfea48e74f7116b724d0887851f57255eb7b.png

Taking the infimum,

measure_theory_4bb390ed9e4617b6e704722ebe553503be193498.png

The Lebesgue measure is then defined on the Lebesgue sigma-algebra, which is the collection of all the sets measure_theory_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png which satisfy the condition that, for every measure_theory_8ce3a5b1041a76dd221dc188f1eac461178977ab.png

measure_theory_ecdfc5ea1fc4dc4ad26cf520ea7c633eaa0b90cd.png

For any set in the Lebesgue sigma-algrebra, its Lebesgue measure is given by its Lebesgue outer measure measure_theory_0667014a76698d6d5aea1a8fe8a96a0f87e62acd.png.

IMPORTANT!!! This is not necessarily related to the Lebesgue integral! It CAN be be, but the integral is more general than JUST over some Lebesgue measure.

Intuition

  • First part of definition states that the subset measure_theory_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png is reduced to its outer measure by coverage by sets of closed intervals
  • Each set of intervals measure_theory_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png covers measure_theory_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png in the sense that when the intervals are combined together by union, they contain measure_theory_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png
  • Total length of any covering interval set can easily overestimate the measure of measure_theory_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png, because measure_theory_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png is a subset of the union of the intervals, and so the intervals include points which are not in measure_theory_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png

Lebesgue outer measure emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fit measure_theory_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png most tightly and do not overlap.

In my own words: Lebesgue outer measure is smallest sum of the lengths of subintervals measure_theory_1c9ba208f7b3c6266197ccf6ed139f510737c985.png s.t. the union of these subintervals measure_theory_1c9ba208f7b3c6266197ccf6ed139f510737c985.png completely "covers" (i.e. are equivalent to) measure_theory_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png.

If you take an a real interval measure_theory_7fffadbade889774e19af9f1f8c0b7927e16f3d7.png, then the Lebesge outer measure is simply measure_theory_718d86e72a0189b0ec0bf9d114c6f5b030fdf1a0.png.

Properties

Notation

  • For measure_theory_f0abf6ff32d5d2aef08884d47da0c496d0559dc7.png and measure_theory_ebdcfdbbd4a497c69547e23cc3342abbf2127bfe.png, we let

    measure_theory_76edd8e202c9edf57650530f93bd1f552390296b.png

Stuff

The collection of Lebesgue measurable sets is a sigma-algebra.

  1. Easy to see measure_theory_2d52d323d5349e9589bfd27e9910b1ca222e825e.png is in this collection:

    measure_theory_fd41fab747604ca0091c334b0e808c4f44dcb67e.png

  2. Closed under complements is clear: let measure_theory_ebdcfdbbd4a497c69547e23cc3342abbf2127bfe.png be Lebesgue measurable, then

    measure_theory_85ac846ff1a6abdf811d05ac4443c0de8ed4a0d9.png

    hence this is also true for measure_theory_ce2bf989a80e4f2fc93924c52bbd66faf039e5e3.png, and so measure_theory_ce2bf989a80e4f2fc93924c52bbd66faf039e5e3.png is Lebesgue measurable.

  3. Closed under countable unions:
    • Finite case: measure_theory_ac8404cc50f2a0b25fcd142bf3dfc3d7417f1ae6.png. Consider measure_theory_dbc574099f87d7a3cc3ca9a00b83d0db5b65879e.png both Lebesgue measurable and some set measure_theory_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png. Since measure_theory_79463e3ac84fc19e2ffdf1cc595c89299b9c5ea1.png is L. measurable:

      measure_theory_9f79f26667683391ecf923f4934e7dd59f03276e.png

      Since measure_theory_d628cc96f9875171d7e1e6e93e1912621b319d2b.png is L. measurable:

      measure_theory_0c58c0d9cbf98a61ff951c96d698fbc15263035a.png

      which allows us to rewrite the above equation for measure_theory_b8f35b18d4bbee98d7c6085e6cb2fc256a59474a.png:

      measure_theory_5c79e6410ac355e286078924f520895769f08f0b.png

      Observe that

      measure_theory_c7dc8716c45694fde2332a966dd8b86fe94d79bd.png

      By subadditivity:

      measure_theory_fb5920d5f9c16caddaa91c9ba9b466ce4e95111c.png

      Hence,

      measure_theory_5e0fbae579e86e372cca5f3fe387bb6847ef1d10.png

      Then this follows for all finite cases by induction.

    • Countable disjoint case: Let measure_theory_c960b38a941a8adb330a12e483a915b6593d9b9f.png, and measure_theory_f9bf6d664d9d2a949a6d9944de07b272a08dcb57.png. Further, let measure_theory_f1337b4a7dc6caba892f11d3aaf147644f0db57c.png.

      measure_theory_0bf8d9a3d597ef434b955e2cf0b6d288ac8dd87c.png

      Hence measure_theory_882e00f5e074530bdb333505df45e42014fea17e.png is L. measurable. Thus,

      measure_theory_bc7688956e2714517022a1b7502dc085cef9e740.png

      Since the measure_theory_f8b648069010a70233704a414386e26e49992dcb.png are disjoint measure_theory_87bc1fa0532c58f11f6ba1a26b6eac9c1fd6a5e0.png and measure_theory_cffe8f7af20d2e36e2e9c06bf3253b227bb1bbec.png:

      measure_theory_7e1ab19e7f66a8d9b107334ec7f6d46e5524f660.png

      Let measure_theory_2335130676f3370d6707f80cef5a7a5e7a99fa14.png and note that measure_theory_aad74bb931b1eb41ee9e19875f442c8c88236534.png. Thus, by indiction

      measure_theory_4e1628b32e03b0a02f416ae55f497010a412cae0.png

      Thus,

      measure_theory_9ede2110718da2cd14a82e05434b83153776e751.png

      Taking measure_theory_2531823f4e49c847dd9a34d97de0ea57802b96fa.png:

      measure_theory_2ea66426d4d10729cdbe2802990d6b171170576f.png

      Thus, measure_theory_c59b765a1913221aa8ecf9a72e6871e304af54f7.png is L. measurable if the measure_theory_f8b648069010a70233704a414386e26e49992dcb.png are disjoint and L. measurable!

    • Countable (not-necessarily-disjoint) case: If measure_theory_f8b648069010a70233704a414386e26e49992dcb.png are not disjoint, let measure_theory_1b95f5d6fd7a442056ee96105a1adcf215c6091e.png and let measure_theory_2324b0a2bc1edf913d30da2073e2732662e4525b.png, which gives a sequence of disjoint sets, hence the above proof applies.

Every open interval is Lebesgue measurable, and the Borel sigma-algebra is a subset of the sigma-algebra of Lebesgue measurable sets.

Want to prove measurability of intervals of the form measure_theory_6c8de0d8619ed05adf34e70fae9ab458f2ac1e34.png.

Idea:

  1. split any set measure_theory_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png into the left and right part
  2. split any cover in the same way
  3. extend covers by measure_theory_b9af43262da05c2b97d226fc619588b3475bb018.png to make them open

measure_theory_17b98d9bdd147be45388776ddcfb2cfb3a70c261.png is a measure space, and for al intervals measure_theory_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png, the measure is the length.

Cantor set

Define

measure_theory_cc3c0c205e952f8bcbd5ed52257b4251f79042e9.png

For measure_theory_84feeb83da8919c3f995550cdfb7e928ee8b390d.png, with measure_theory_134bf3e1494433568e7251f90dd66aef06793a5e.png being identity, and

measure_theory_61c52329262caac8c8b59ee3d3c0ca094ac58b36.png

Let measure_theory_2afd7752814da142712565c8b8f65ca698bbbf44.png and measure_theory_fa64173e3c0b35b7290c9bfdbdb747ae390936eb.png. Then the Cantor set is defined

measure_theory_bc9777ff3aa2bea75beea6d0a32236a76e737a0d.png

The Cantor set has a Lebesgue measure zero.

We make the following observations:

  • Scaled and shifted closed sets are closed
  • measure_theory_049620bed349a8deabe58aa25ed270a33ddb7c9d.png is a finite union of closed intervals and so is in the Borel sigma-algebra
  • σ-algebras are closed under countable intersections, hence Cantor set is in the Borel σ-algebra
  • Finally, Borel σ-algebra is a subset of Lebesgue measurable sets, hence the Cantor set is Lebesuge measurable!

Since Lebesgue measure satisfy measure_theory_20bb6f29a7de923a850f8530c284a7c9ece2c3de.png for any Lebesgue measurable set measure_theory_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png with finite measure and any measure_theory_f0abf6ff32d5d2aef08884d47da0c496d0559dc7.png with measure_theory_9e09e0a27e8c3c465eb0df506b5c33273065229d.png. Since Lebesgue measure is subadditive, we have for any measure_theory_fd6c30cd4dc619de3bdc2b9705dcd5acce7b1ca3.png

measure_theory_cfb0ae5f7c102ebfa94e8c86b99082ec23efdac9.png

Since measure_theory_c3bda9bc2f2044313fea326fe5d355ec649e7a21.png, by induction, it follows that

measure_theory_d52d4a946528ee138db587a0ea34081112ca72b5.png

Taking the infimum of over measure_theory_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png, we have that the Cantor set has measure zero:

measure_theory_f006726ea56afca816315eb352999759c613e539.png

Cardinality of the Cantor set

Let measure_theory_31a3d0292a794fb3cbea1fbccc31638235a1ac7d.png.

The terniary expansion is a sequence measure_theory_ae5d14e52cdcb2669ad4a6708ec8ce1f81d78365.png with measure_theory_73d2d351f077eff604cc3b299fda8b3aaa81c1ea.png such that

measure_theory_682bd1df6b1a7e092bb7ba015a3c8f858265f985.png

The Cantor set measure_theory_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png is uncountable.

We observe that if the first measure_theory_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png elements of the expansion for measure_theory_7a84c9a383f9772338016d101ccc096be06af784.png are in measure_theory_517f20da743e0ea2449ea07283c538de682da3ce.png, then measure_theory_2caff65cd664f7c1130a29b01bb5dcbd42429d31.png. But importantly, observe that some numbers have more than one terniary expansion, i.e.

measure_theory_2564684bd1570318c667db4a628802e94946835d.png

in the terniary expansion. One can show that a number measure_theory_a75b7783a1f07c2000e3a9ef2d944140b38f6435.png if and only if measure_theory_7a84c9a383f9772338016d101ccc096be06af784.png has a terniary expansion with no 1 digits. Hence, the Cantor set measure_theory_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png is uncountable!

One can see that measure_theory_a75b7783a1f07c2000e3a9ef2d944140b38f6435.png if and only if terniary expansion with no 1 digits, since such an measure_theory_7a84c9a383f9772338016d101ccc096be06af784.png would land in the "gaps" created by the construction of the Cantor set.

Uncountable Lebesgue measurable set

There exists uncountable Lebesgue measurable sets.

Menger sponge

Vitali sets

Let measure_theory_d04760174a49c59de7013fce311eeadf7c9a12ff.png if and only if measure_theory_9138ca07c3d6d0bfb6bba6849ccfe54a8131a66f.png.

  • There are uncountable many equivalence classes, with each equivalence class being countable (as a set).
  • By axiom of choice, we can pick one element from each equivalence class.
  • Can assume each representative picked is in measure_theory_5cb819dbdaa11557a460fe04a5ef95eede596daa.png, and this set we denote measure_theory_3d136f0fc4860468633907421c098b9feb0eef24.png

Suppose, for the sake of contradiction, that measure_theory_3d136f0fc4860468633907421c098b9feb0eef24.png is measurable.

Observe if measure_theory_31a3d0292a794fb3cbea1fbccc31638235a1ac7d.png, then there is a measure_theory_753216268469ee3dc1c55ac2efc7742109171a85.png and measure_theory_fd09f917138239c467ef35b147b0da01a59d8f60.png s.t. measure_theory_5c607d4a61420e23bc1352b1fee89042ab1287d1.png, i.e.

measure_theory_ed65e0f0bd36746ac8d85652b1d343faa1e807d3.png

Then, by countable additivity

measure_theory_996fc9df8e3228e7c22b8b552b03425361c35cd9.png

where we've used

measure_theory_7adbc9e1c07e35d08aae72d9be80f660b407c729.png

Hence, we have our contradiction and so this set, the Vitali set, is not measurable!

There exists a subset of measure_theory_492d525117d0dcc93d066c8759f46b98cf9980ca.png that is not measurable wrt. Lebesgue measure.

Lebesgue Integral

The Lebesgue integral of a function measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png over a measure space measure_theory_85836c3c18ee764fbdef801c3631f37f2456dcf7.png is written

measure_theory_dd35ce19f3448e5b556b5e054674644f232e5ce4.png

which means we're taking the integral wrt. the measure measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png.

riemann_vs_lebesgue_integral.png

Special case: non-negative real-valued function

Suppose that measure_theory_95582f089b0a75752fa0a50d824485f0482bd980.png is a non-negative real-valued function.

Using the "partitioning of range of measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png" philosophy, the integral of measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png should be the sum over measure_theory_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png of the elementary area contained in the thin horizontal strip between measure_theory_11c971d161e3a32855dca081c99ac51d239a1115.png and measure_theory_93479b23df7cf877d3bdecca9b3a4207e181399f.png, which is just

measure_theory_64561d2e90b7acf7a6d1bb28653ebf5d25940d1e.png

Letting

measure_theory_8b6c5d7fb3a7335c2a880b0962a0e76f46126e59.png

The Lebesgue integral of measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png is then defined by

measure_theory_d2cc8b987115b5eaca1757ef5cf9bf559aa2861b.png

where the integral on the right is an ordinary improper Riemann integral. For the set of measurable functions, this defines the Lebesgue integral.

Radon measure

  • Hard to find a good notion of measure on a topological space that is compatible with the topology in some sense
  • One way is to define a measure on the Borel set of the topological space

Let measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png be a measure on the sigma-algebra of Borel sets of a Hausdorff topological space measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

  • measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png is called inner regular or tight if, for any Borel set measure_theory_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png, measure_theory_db375b73c0e07268ccfddecb004db6b906fb51ad.png is the supremum of measure_theory_7c257de45927e5350f18c475d8566c8dc366c1d4.png over all compact subsets of measure_theory_1641d18cc980f8db14cdff95d7417a8526eef446.png of measure_theory_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png
  • measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png is called outer regular if, for any Borel set measure_theory_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png, measure_theory_db375b73c0e07268ccfddecb004db6b906fb51ad.png is the infimum of measure_theory_1d9db1d5a726ac7f5bb690e7ad6c7c4bd3e9e614.png over all open sets measure_theory_da9cb51849b13210fa778a80b9907f86fe90c379.png containing measure_theory_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png
  • measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png is called locally finite if every point of measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png has a neighborhood measure_theory_da9cb51849b13210fa778a80b9907f86fe90c379.png for which measure_theory_1d9db1d5a726ac7f5bb690e7ad6c7c4bd3e9e614.png is finite (if measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png is locally finite, then it follows that measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png is finite on compact sets)

The measure measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png is called a Radon measure if it is inner regular and locally finite.

Suppose measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png and measure_theory_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png are two measure_theory_026b58b7d0ddfb732bc0a0630d01a61cc001cc27.png measures on a measures on a measurable space measure_theory_5537083cdab42c7108d37538c78dc872ac285109.png and measure_theory_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png is absolutely continuous wrt. measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png.

Then there exists a non-negative, measurable function measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png on measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png such that

measure_theory_336455e85c2e8723cd7d75560079d025a89ef0b3.png

The function measure_theory_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png is called the density or Radon-Nikodym derivative of measure_theory_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png wrt. measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png.

If a Radon-Nikodym derivative of measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png wrt. measure_theory_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png exists, then measure_theory_a881862122dfd4df3ea0a0c00835db8993b27f11.png denotes the equivalence class of measurable functions that are Radon-Nikodym derivatives of measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png wrt. measure_theory_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png.

measure_theory_0e82c02fca034c87940bd23eaa53905cabec0cb3.png is often used to denote measure_theory_c53c85486513e0d0611326c3980dc257a7d2631e.png, i.e. measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png is just in the equivalence class of measurable functions such that this is the case.

This comes from the fact that we have

measure_theory_39673294ef7c2a1240b0b87ee9435ec583efd6f9.png

Suppose measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png and measure_theory_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png are Radon-Nikodym derivatives of measure_theory_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png wrt. measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png iff measure_theory_e24678818de4aa54c2bb5b2659ff250295cccff0.png.

The δ measure cannot have a Radon-Nikodym derivative since integrating measure_theory_18f46e29ae5df6d58883632ac9a73c6a3ea7e102.png gives us zero for all measurable functions.

Continuity of measure

Suppose measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png and measure_theory_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png are two sigma-finite measures on a measure space measure_theory_f15b78d0d3d5c4397bb64f0f460372309bae5e57.png.

Then we say that measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png is absolutely continuous wrt. measure_theory_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png if

measure_theory_a7b0b8568bae9b6a2e1f2e6e474c2ad3166ab962.png

We say that measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png and measure_theory_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png are equivalent if each measure is absolutely continuous wrt. to the other.

Density

Suppose measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png and measure_theory_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png are two sigma-finite measures on a measure space measure_theory_f15b78d0d3d5c4397bb64f0f460372309bae5e57.png and that measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png is absolutely continuous wrt. measure_theory_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png. Then there exists a non-negative, measurable function measure_theory_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png on measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png such that

measure_theory_d9583a6befa91297a06b1a5afb9d158ceb37de3c.png

Measure-preserving transformation

measure_theory_61e5589f0f948fdfe5922eb14ba15d50dc67c567.png is a measure-preserving transformation is a transformation on the measure-space measure_theory_85836c3c18ee764fbdef801c3631f37f2456dcf7.png if

measure_theory_736e98111fbdfac22260d34ea68d36fefccce424.png

Measure

A measure on a set is a systematic way of defining a number to each subset of that set, intuitively interpreted as size.

In this sense, a measure is a generalization of the concepts of length, area, volume, etc.

Formally, let measure_theory_37d3d2a40172b2cf7b0e8ea91ec033c3e1f9ecbd.png be a measure_theory_dcb897d0137330758a9675e99ee9ec7f93c4d742.png of subsets of measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

Suppose measure_theory_bf6e9a6f76768da13416c6597a7f3fe1839ba3f4.png is a function. Then measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png is a measure if

  1. measure_theory_a1a5a70f855c26c9d1ddb6800e67fef382077d41.png
  2. Whenever measure_theory_fdce91383bd04dc87d59be7e69dd0eaaedb73bac.png are pairwise disjoint subsets of measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png in measure_theory_37d3d2a40172b2cf7b0e8ea91ec033c3e1f9ecbd.png, then

    measure_theory_9b164cacdbdd577cae6d52ddd94d9f7686993b30.png

Properties

Let measure_theory_5e9da78c6f93a1fa8bfcb8a8115e7fa75932e95f.png be a measure space, and measure_theory_abe2f13bb5f46d273b4c9f50f83570f45fbdaca4.png such that measure_theory_1ead3bb1976ff41be59bbf1a9718e3cb7944be08.png.

Then measure_theory_3952d0ddd0979e0ba907cb82d34597acb02588b3.png.

Let

measure_theory_89e7a9f4b0b90d8666e7c5afe6ee6cd62810db4a.png

Then measure_theory_5711476e727600d517690d425cd46c540b6eb732.png, and by finite additivity property of a measure:

measure_theory_9afc6b29dabe9c7f55ab72974d46b447708a06eb.png

since measure_theory_b281b2a5ad3f78ab6e9fd17682d644e5823a1065.png by definition of a measure.

If measure_theory_fdce91383bd04dc87d59be7e69dd0eaaedb73bac.png are measure_theory_534c9a75a46d2155b39f712071ce1c5782a66e4f.png subsets of measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, then

measure_theory_43211515a345b5ed8137990db2a8021e5eacd717.png

We know for a sequence of disjoint sets measure_theory_88720d0c7e591e4ffac9d00f9cd6c8d344404537.png we have

measure_theory_e6a877c40ef8e86d35448dd4483b64373a8c1826.png

So we just let

measure_theory_81c2503b22c3db060424e0695f63fd7fd97a5297.png

Then,

measure_theory_757c496acdfc83a123634a4e0455a7fede784447.png

Thus,

measure_theory_67b27fe079b123065dfc0e609740cbf0ab4effff.png

Concluding our proof!

Let measure_theory_f61a8dcd0e109eae5ea40dad40159fe901ee7675.png be an increasing sequence of measurable sets.

Then

measure_theory_722d22656dab8895ada6461ad060980c9eac5e55.png

Observe

measure_theory_d639d81daaf0bdeef32bccf84f4d144243556029.png

then by monotonicity of the measure we have

measure_theory_7200a4c17eab86d9dad10a976bf943051ca5aeeb.png

Which is true for all measure_theory_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png, and so

measure_theory_de2e166f7f20a91c83e286bf2e34c5cfd9ac44cf.png

as wanted.

Let measure_theory_fcf78e9a378273e60b7fb562857d2d5b311e695d.png be sets from some measure_theory_dcb897d0137330758a9675e99ee9ec7f93c4d742.png measure_theory_37d3d2a40172b2cf7b0e8ea91ec033c3e1f9ecbd.png.

If measure_theory_70779f277d948d64a0ea6b5cae0187175ec54dd1.png, then

measure_theory_3e0439d3ca7e6a83d5264bfd36b1df519148bd98.png

Examples of measures

Let

  • measure_theory_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png be a space
  • measure_theory_83f3859a8adf76ea46b6a2652a5e35a1254780b5.png

The δ-measure (at measure_theory_3778a663528a3e4db5ce811ed0c6cd9f5831579f.png) is

measure_theory_179e66912a4100b74e2892a63b1fd1739dedfe76.png

Sigma-algebra

Definition

Let measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be some set, and let measure_theory_6f356776de41b193d9421916df1afc027221c49d.png be its power set. Then the subset measure_theory_560131fab33600e7156a7fdb2a816b224a46e25c.png is a called a σ-algebra on measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png if it satisfies the following three properties:

  1. measure_theory_868d3f8a55a0dc55d25635cc1f361377469deb51.png
  2. measure_theory_017948b866be67b1a8e56a6b5f8848f823410c34.png is closed under complement: if measure_theory_d74935f39042b03a6ff552bcd4802908fffe3d5d.png
  3. measure_theory_017948b866be67b1a8e56a6b5f8848f823410c34.png is closed under countable unions: if measure_theory_c1dc820d66c3a8e96360ca48754b7b9a39628344.png

These properties also imply the following:

  • measure_theory_3416854bb7897915e8900878d7fdd4ce77fab8f1.png
  • measure_theory_017948b866be67b1a8e56a6b5f8848f823410c34.png is closed under countable intersections: if measure_theory_8b7e76de03d521b4dd4cbba867549ef707be4cd1.png

Generated σ-algebras

Given a space measure_theory_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png and a collection of subsets measure_theory_faf5c8be75250ee9d78666266f0c497eede494d2.png, the σ-algebra generated by measure_theory_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png, denoted measure_theory_da7760c8308f40b9f0c18c295c53ed110a604871.png, is defined to be the intersection of all σ-algebras on measure_theory_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png that contain measure_theory_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png, i.e.

measure_theory_8af43beb60270a27f96520128b54afbfa22add5a.png

where

measure_theory_2967824f4ec19fc40b5da14ce1ceed6019856c6f.png

Let measure_theory_5537083cdab42c7108d37538c78dc872ac285109.png be a measurable space and measure_theory_b4e203d9c8380478a39f56bc5bbf9309bd234c48.png a function from some space measure_theory_017948b866be67b1a8e56a6b5f8848f823410c34.png to measure_theory_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png.

The σ-algebra generated by measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png is

measure_theory_3678832fb7143efebde155e9e8328a5ac538ba9a.png

Observe that though this is similar to σ-algebra generated by MEASURABLE function, the definition differs in a sense that the preimage does not have to be measurable. In particular, the σ-algebra generated by a measurable function can be defined as above, where measure_theory_a219cdba3f608b95560d9bcca1ff2a7af7f8346e.png is measurable by definition of measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png being a measurable function, hence corresponding exactly to the other definition.

Let measure_theory_36500b4957975900bceded963ae04e7c6e611b2d.png and measure_theory_3b6a97c8a3308313c250bb1bd7659ca00d951f62.png be measure spaces and measure_theory_97667d9428e95e5a339eeaccd10a267f23ae3539.png a measurable function.

The σ-algebra generated by measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png is

measure_theory_9b4e69b1552d919f12b0aad81c135d47d6dbcf93.png

Let measure_theory_bb22d7e3135e825adc3652a3a51866d686beb4d3.png be a probability space and measure_theory_c6c5ef77ef86935c94680ab75e763a010e80e2e4.png a random variable.

The σ-algebra generated by measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is

measure_theory_2e7362f873f0a61c2e804169b169f0bbc4c6da3b.png

Let measure_theory_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png be a space.

If measure_theory_7b99a169302e4b9580d30c1668b00770fc3812da.png is a collection of σ-algebras, then measure_theory_f5ed1454d87360435feb23c6af2ae17436c05735.png is also a σ-algebra.

σ-finite

A measure or premeasure space measure_theory_b88799d5ec72ed146276eedc13cdcd9204986c5c.png is finite if measure_theory_1943b6e3e91858f21699123c557fbfb715e56c55.png.

A measure measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png on a measure space measure_theory_8c9410961da365661c5bac7225665630a902ea74.png is said to be sigma-finite if measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png can be written as a countable union of measurable sets of finite measure.

Example: counting measure on uncountable set is not σ-finite

Let measure_theory_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png be a space.

The counting measure is defined to be measure_theory_231cf5a52a5f6e576a4f7f7ee299e4bd28aa6cbb.png such that

measure_theory_5cbf7d69200fa8542f6ed5c299b3d3f25b8e0333.png

On any uncountable set, the counting measure is not σ-finite, since if a set has finite counting measure it has countably many elements, and a countable union of finite sets is countable.

Properties

Let measure_theory_37d3d2a40172b2cf7b0e8ea91ec033c3e1f9ecbd.png be a measure_theory_dcb897d0137330758a9675e99ee9ec7f93c4d742.png of subsets of a set measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png. Then

  1. measure_theory_5cfbe30558b4059333a131f5d65765edd2c8efbe.png
  2. If measure_theory_40e086907859cdbe9f84b4956e34e681bf90870a.png, then

    measure_theory_9d063482329f3488fa029c5edff13a77d2d8f3ca.png

  3. If measure_theory_abe2f13bb5f46d273b4c9f50f83570f45fbdaca4.png then measure_theory_fdcb518aa4a84806c2d08c29bbc9298ccf24f6a4.png

Borel sigma-algebra

Any set in a topological space that can be formed from the open sets through the operations of:

  • countable union
  • countable intersection
  • complement

is called a Borel set.

Thus, for some topological space measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, the collection of all Borel sets on measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png forms a σ-algebra, called the Borel algebra or Borel σ-algebra .

More compactly, the Borel σ-algebra on measure_theory_492d525117d0dcc93d066c8759f46b98cf9980ca.png is

measure_theory_6b37501a8387bfd2c7a903b29ea53a0513089e92.png

where measure_theory_97f35052cf30dc8e2b4bfe32c1462a02476ead1a.png is the σ-algebra generated by the standard topology on measure_theory_492d525117d0dcc93d066c8759f46b98cf9980ca.png.

Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space.

Any measure defined on the Borel sets is called a Borel measure.

Lebesgue sigma-algebra

Basically the same as the Borel sigma-algebra but the Lebesgue sigma-algebra forms a complete measure.

Note to self

Suppose we have a Lebesgue mesaure on the real line, with measure space measure_theory_02b3b2e9358326ceefbd0f9e5ac4014d40fba374.png.

Suppose that measure_theory_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is non-measurable subset of the real line, such as the Vitali set. Then the measure_theory_4a04566b671f768402c9b91f57707ea1523d782b.png measure of measure_theory_9574d40571bf38a0136479fd0d504d2e719c41f2.png is not defined, but

measure_theory_eed109b95a9038895a22ec05946d227a4bb22b20.png

and this larger set ( measure_theory_3ff49ed7ddd4f67d576c9ce3905bdbe1bfee4047.png ) does have measure_theory_4a04566b671f768402c9b91f57707ea1523d782b.png measure zero, i.e. it's not complete !

Motivation

Suppose we have constructed Lebesgue measure on the real line: denote this measure space by measure_theory_02b3b2e9358326ceefbd0f9e5ac4014d40fba374.png. We now wish to construct some two-dimensional Lebesgue measure measure_theory_4a04566b671f768402c9b91f57707ea1523d782b.png on the plane measure_theory_e4f375c26796781f71b7ae3026445db617a6e78b.png as a product measure.

Naïvely, we could take the sigma-algebra on measure_theory_e4f375c26796781f71b7ae3026445db617a6e78b.png to be measure_theory_f511b32cdc5972418b6443b060202d9dc94e42c2.png, the smallest sigma-algebra containing all measureable "rectangles" measure_theory_5a83e273444069ea4d94185dfcb1f38b0e9b2413.png for measure_theory_34fd1ec18df7dc8e4893bc7b138da6231940004d.png.

While this approach does define a measure space, it has a flaw: since every singleton set has one-dimensional Lebesgue measure zero,

measure_theory_2fd48313f6009dfc982b60c4792f3d2667872916.png

for any subset of measure_theory_492d525117d0dcc93d066c8759f46b98cf9980ca.png.

What follows is the important part!

However, suppose that measure_theory_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is non-measureable subset of the real line, such as the Vitali set. Then the measure_theory_4a04566b671f768402c9b91f57707ea1523d782b.png measure of measure_theory_9574d40571bf38a0136479fd0d504d2e719c41f2.png is not defined (since we just supposed that measure_theory_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is non-measurable), but

measure_theory_eed109b95a9038895a22ec05946d227a4bb22b20.png

and this larger set ( measure_theory_3ff49ed7ddd4f67d576c9ce3905bdbe1bfee4047.png ) does have measure_theory_4a04566b671f768402c9b91f57707ea1523d782b.png measure zero, i.e. it's not complete !

Construction

Given a (possible incomplete) measure space measure_theory_85836c3c18ee764fbdef801c3631f37f2456dcf7.png, there is an extension measure_theory_af9d9087db2dc793391a11589265145931b00dc3.png of this measure space that is complete .

The smallest such extension (i.e. the smallest sigma-algebra measure_theory_f992e42c2c120dd425fbadf62f2573acb2187637.png ) is called the completion of the measure space.

It can be constructed as follows:

  • Let measure_theory_ca2bcc80309e331f71dd2c1e1f4788cbc048c242.png be the set of all measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png measure zero subsets of measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png (intuitively, those elements of measure_theory_ca2bcc80309e331f71dd2c1e1f4788cbc048c242.png that are not already in measure_theory_017948b866be67b1a8e56a6b5f8848f823410c34.png are the ones preventing completeness from holding true)
  • Let measure_theory_f992e42c2c120dd425fbadf62f2573acb2187637.png be the sigma-algebra generated by measure_theory_017948b866be67b1a8e56a6b5f8848f823410c34.png and measure_theory_ca2bcc80309e331f71dd2c1e1f4788cbc048c242.png (i.e. the smallest sigma-algreba that contains every element of measure_theory_017948b866be67b1a8e56a6b5f8848f823410c34.png and of measure_theory_ca2bcc80309e331f71dd2c1e1f4788cbc048c242.png)
  • measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png has an extension to measure_theory_f992e42c2c120dd425fbadf62f2573acb2187637.png (which is unique if measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png is sigma-finite), called the outer measure of measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png, given by the infimum

measure_theory_d39cc89a7d78a75ce7924aec030fcaf2e2bf7c4e.png

Then measure_theory_af9d9087db2dc793391a11589265145931b00dc3.png is a complete measure space, and is the completion of measure_theory_85836c3c18ee764fbdef801c3631f37f2456dcf7.png.

What we're saying here is:

  • For the "multi-dimensional" case we need to take into account the zero-elements in the resulting sigma-algebra due the product between the 1D zero-element and some element NOT in our original sigma-algebra
  • The above point means that we do NOT necessarily get completeness, despite the sigma-algebras defined on the sets individually prior to taking the Cartesian product being complete
  • To "fix" this, we construct a outer measure measure_theory_dc3dda5fbf9d111629db86b3e3690b394b7d7d32.png on the sigma-algebra where we have included all those zero-elements which are "missed" by the naïve approach, measure_theory_f992e42c2c120dd425fbadf62f2573acb2187637.png

Measurable functions

Let measure_theory_12350c3ffd3bf17592d07e0f060019df0069b43d.png and measure_theory_fb1048c2bafc904db7b4a7e086c9c92c3ec708cf.png be measurable spaces.

A function measure_theory_8fd511771ed24948876570c0e0ef76a099b637c2.png is a measurable function if

measure_theory_358ed0b6404dba69544527275507b389ed84a7ce.png

where measure_theory_5dc6ef2a80b525f99fc8d6f112a0268cc5f1e07a.png denotes the preimage of the measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png for the measurable set measure_theory_67ff80d80c3d1ff30b1b4cd454913dfe50bbe266.png.

Let measure_theory_e82bed926aa8ba1bb7f4e5dc003ea7e7808b8eac.png.

We define the indicator function of measure_theory_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png to be the function measure_theory_299161fc0d9ce277bc873b312947918614e996b1.png given by

measure_theory_b63922cf147c6f6adf1cf0ee0e1f58895d042992.png

Let measure_theory_e82bed926aa8ba1bb7f4e5dc003ea7e7808b8eac.png. Then measure_theory_8cb94a32b93fca2e76396dd9b96c4be2760eacba.png is measurable if and only if measure_theory_41e6ce2ab80715fe484d74efc80263e1528225f7.png.

Let measure_theory_5e9da78c6f93a1fa8bfcb8a8115e7fa75932e95f.png be a measure space or a probability space.

Let measure_theory_60e3ddf377ca917dec9214abeacf0b3e56e26509.png be a sequence of measurable functions.

  1. For each measure_theory_f56b556b5aef28b769c52c4c0bf2d3213e67eb2e.png, the function measure_theory_237283230fd59bd9bf8a3dc86b288c5c488b5073.png is measurable
  2. The function measure_theory_08d958f4d5dc1e037354f523f3b622d7f6952088.png is measurable
  3. Thus, if measure_theory_617b87515c58d75e4faeff2a8dbc4b795e512b7c.png converge pointwise, measure_theory_19caad1724d104f0615e2ae05e4e80644d6a50cb.png is measurable.

Let measure_theory_5d83d81f30271966c4509b4a0242f6209b45d981.png be a measurable space, and let measure_theory_8f67a6d9738808f86dd719329c85c73be4b2def2.png.

The following statements are equivalent:

  1. measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png is measurable.
  2. measure_theory_65e3e176dfbaca1b9755de69c95c08c4f8066aea.png we have measure_theory_79c16edbd9924bda4401a75bd2f6c7bbef108713.png.
  3. measure_theory_aefff55fb5cfea5a66f0b5bfcadef4084f4836fa.png we have measure_theory_4363db8a524ebe50d17255159909837eed47a17b.png.
  4. measure_theory_3792952f68fe8e824151363ea94775ade04ba253.png we have measure_theory_779aa2a8bc35ddb76aec458bf125b65ce1fe2962.png.
  5. measure_theory_aefff55fb5cfea5a66f0b5bfcadef4084f4836fa.png we have measure_theory_2a387c68f5ae3811002768439416a721a163a26e.png.

A function measure_theory_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png is measurable if

measure_theory_6ef7557498724984d9ec5d20457d1dcecdc074bb.png

We also observe that by Proposition proposition:equivalent-statements-to-being-a-measurable-function, it's sufficient to prove

measure_theory_00e8aab7eb229e4751749cad04c5474bbee82c78.png

so that's what we set out to do.

For measure_theory_f56b556b5aef28b769c52c4c0bf2d3213e67eb2e.png and measure_theory_4c91032750887d8dca229b906727af58e9268232.png, consider the following equivalent statements:

measure_theory_633cbbaa1591af3ab1ca7af21bb7213a9773a0bf.png

Thus,

measure_theory_dc0cc083772e454b4cbdc9b46db9c8c84a638fff.png

so

measure_theory_92db937371a71c28f3ece4004bcf3c23b7a0cc7d.png

Recall that for each measure_theory_4c91032750887d8dca229b906727af58e9268232.png, the sequence measure_theory_e575f096f16dc4b3fed3e18bc8af9e7b6177d3fd.png is an increasing sequence in measure_theory_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png. Therefore, similarily, the following are equivalent:

measure_theory_c607c1f7340bf1a1c0e3953344ae56b783e7f9fc.png

Thus,

measure_theory_ece017521daa50434be44d77ccf9307323002c72.png

Hence,

measure_theory_b20ad8d37b8b1e98fdd1a5209679122c84ee7d34.png

concluding our proof!

Basically says the same as Prop. proposition:limits-of-measurable-functions-are-measurable, but a bit more "concrete".

Let measure_theory_37d3d2a40172b2cf7b0e8ea91ec033c3e1f9ecbd.png be a measure_theory_dcb897d0137330758a9675e99ee9ec7f93c4d742.png of subsets of a set measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, and let measure_theory_eaa92f564c5fc0aad26c09e130bd4ab63072dde7.png with measure_theory_963ce6c997d781c06cfcdd821a0b72c33c5dd6c4.png be a sequence of measurable functions.

Furthermore, let

measure_theory_1002569af12a24fed9f2148e76017b49ff6dc95f.png

Then measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png is a measurable function.

Simple functions

Let measure_theory_37d3d2a40172b2cf7b0e8ea91ec033c3e1f9ecbd.png be a measure_theory_dcb897d0137330758a9675e99ee9ec7f93c4d742.png of subsets of a set measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

A function measure_theory_3e793fe48d1151fd67e7f9191bcc80d5a70b99d7.png is called a simple function if

  • it is measurable
  • only takes a finite number of values

Let measure_theory_37d3d2a40172b2cf7b0e8ea91ec033c3e1f9ecbd.png be a measure_theory_dcb897d0137330758a9675e99ee9ec7f93c4d742.png of subsets of a set measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

Let measure_theory_1c937903845a2d88a7908e3b6cbc0c2c272be0d6.png be a nonnegative measurable function.

Then there exists a sequence measure_theory_6f800e8020f8ca02c4e110fc5eb1ac0e8111039c.png of simple functions such that

  1. measure_theory_160114398c4d8ee190226b3b938c8cefc367ce54.png for all measure_theory_4c91032750887d8dca229b906727af58e9268232.png
  2. Converges to measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png:

    measure_theory_cce7b26e51bbab99db15793349af8a1dc22daf1a.png

Define a function measure_theory_6026c03e845e7bf21559f8d9abba4def5de3e62f.png as follows. Let

measure_theory_bf0ef70cceaf62d6af7457d7e3f9bfbeb404c1c5.png

and let

measure_theory_334d41b8dc5ea1ed796b4eb5a59099e662ba3690.png

Then the function

measure_theory_7717c3c48e3a17b32a26fbfc4adea2290344b2d5.png

obeys the required properties!

Almost everywhere and almost surely

Let measure_theory_b88799d5ec72ed146276eedc13cdcd9204986c5c.png be a measure or probability space.

Let measure_theory_23726c31cc6c1501e8a079c2ca7b4e5dfde88913.png be a sequence of measurable functions

  1. For each measure_theory_f56b556b5aef28b769c52c4c0bf2d3213e67eb2e.png the function measure_theory_237283230fd59bd9bf8a3dc86b288c5c488b5073.png is measurable
  2. The function measure_theory_08d958f4d5dc1e037354f523f3b622d7f6952088.png is measurable
  3. Thus, if the measure_theory_617b87515c58d75e4faeff2a8dbc4b795e512b7c.png converge pointwise, then measure_theory_19caad1724d104f0615e2ae05e4e80644d6a50cb.png is measurable

Let measure_theory_b88799d5ec72ed146276eedc13cdcd9204986c5c.png be a measure space. Let measure_theory_d21892f3bdfae9ec08a78f3061484c467c1030ec.png be a condition in oe variable.

measure_theory_d21892f3bdfae9ec08a78f3061484c467c1030ec.png holds almost everywhere (a.e.) if

measure_theory_b9900bddf11c330a0935a1fbd46bfee9aef5c645.png

Let measure_theory_bb22d7e3135e825adc3652a3a51866d686beb4d3.png be a probability space and measure_theory_d21892f3bdfae9ec08a78f3061484c467c1030ec.png be a condition in one variable, then measure_theory_d21892f3bdfae9ec08a78f3061484c467c1030ec.png holds almost surely (a.e.) if

measure_theory_1da88559ee59e8d3d149248e1bd5f3744619051e.png

also denoted

measure_theory_29ca3cd94638559873a1fb52d4ffa10211395ebe.png

Let measure_theory_b88799d5ec72ed146276eedc13cdcd9204986c5c.png be a complete measure space.

  1. If measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png is measurable and if measure_theory_c9e063cb396197d3fa3c2e75b8ba069fd908c76c.png a.e. then measure_theory_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png is measurable.
  2. Being equal a.e. is an equivalence relation on measurable functions.

Convergence theorems for nonnegative functions

Problems

Let measure_theory_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png be Lebesgue measurable and measure_theory_714e6649d9a4c2189ca99a2c4afb6c39cb975c3b.png.

Then measure_theory_354922838372720193f9737f191c5f9be3eb221e.png with measure_theory_9c2f16862ccf2498b205c4cb227c1866a266f2c3.png s.t.

measure_theory_02c25b619f3bda1d5678e0261dba10aa308bef34.png

Clearly if measure_theory_354922838372720193f9737f191c5f9be3eb221e.png with measure_theory_9c2f16862ccf2498b205c4cb227c1866a266f2c3.png s.t. measure_theory_4aa5f976dd5dba091cb3772660facc2b8c06741f.png, then

measure_theory_7f919b3b4017edc94535104ac3ee0ab65ef90fd1.png

hence

measure_theory_87954944b7a9f8f45529503d93c5be44093fd735.png

Therefore it's sufficient to prove that if measure_theory_714e6649d9a4c2189ca99a2c4afb6c39cb975c3b.png, then there exists a non-degenerate open interval measure_theory_4dbac29c08ad38dc09a53b1d71656a5e05b88888.png s.t. measure_theory_3455f994faac26b665b526742da557bae0820356.png. (first I said contained in measure_theory_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png, but that is a unecessarily strong statement; if contained then what we want would hold, but what we want does not imply containment).

As we know, for every measure_theory_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png there exists measure_theory_a86b20be0707983bd88fd1bd906b7590af2ba342.png such that measure_theory_f72625aa3eb1cd99c34056b45357f16db9679cca.png and

measure_theory_b052f542af362f40381611b2f7b0be897cda6644.png

Which implies

measure_theory_264aeae94be1ec7d395cd62972901cc6462451b1.png

which implies

measure_theory_cc73ab86e2de70b7a34ff2de0e62afdd452da6db.png

Letting measure_theory_d1b02fae817763a9940d653a2ee018c296e7869b.png, this implies that there exists an open cover measure_theory_2f786c1bec4d4a2f93348c4b1b5a24abff23c0e9.png s.t.

measure_theory_3951a75a279a091d8620230ea84f215b22aa056c.png

and

measure_theory_99c1fc31cca5ec584b7d358d465224d65d9cc34b.png

(this fact that this is true can be seen by considering measure_theory_bdc1a896fbdc18cda32973ef6db7c742948183bd.png for all measure_theory_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png and see that this would imply measure_theory_2a60517b4e6f1a17a8fb849af701255f09e52019.png not being a cover of measure_theory_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png, and if measure_theory_caa0335d690d80d9778508ed66f5e5fbd0f94f26.png, then since measure_theory_e9757201e8419f489084b3e4a19b890ceb96ce26.png there exists a "smaller" cover).

Thus,

measure_theory_04ac8dd4074527e67442f2b9ad6bc6e59dc0e9e9.png

Hence, letting measure_theory_ace2bca881e8ad00894c8bc9404c9d32b19d7e07.png be s.t.

measure_theory_c2cb0b8d08bae60ea9f7a38f117caca277b40a7e.png

we have

measure_theory_14ecf4046b7fc7b80b35697faebf410af007c964.png

as wanted!

measure_theory_0d6385680f9f50537637e277cd21d9caef3c8836.png, we have measure_theory_a63e2e479892e0dd74fed26b36e4c298170e5a07.png for almost every measure_theory_5ae6b36dba66b3d59a4a16a87b3f05f8bdab18de.png if and only if for almost every measure_theory_5ae6b36dba66b3d59a4a16a87b3f05f8bdab18de.png, measure_theory_a63e2e479892e0dd74fed26b36e4c298170e5a07.png for all measure_theory_84feeb83da8919c3f995550cdfb7e928ee8b390d.png.

This is equivalent to saying

measure_theory_ae2b9e0589d969414ff09d9e1459bc43b7c6ef66.png

if and only if

measure_theory_38ce8d8e204dbbefdaff119ee2e876201987d4f2.png

i.e. measure_theory_d305da3d342ec711303e75c9cb51cc99571e2fe0.png is a set of measure zero.

measure_theory_8d1e4cdf6830262655c72e89fe6ada114f49d707.png Then clearly

measure_theory_06d73ebe56d04a0ef543b6c302d6232d1d5d8599.png

by the assumption.

measure_theory_71cec19ca12a2dae45ffb1f0ad093b0ddf05dcf2.png Follows by the same logic:

measure_theory_711ff940bf2e4de3351a13bce9596ed26697a329.png

This concludes our proof.

Integration

Notation

  • We let

    measure_theory_15b03bf5b846962d784e7c60a453ce5392e302d1.png

    where

    measure_theory_9afbcc74e3dafc2754845fb567ef929370d97751.png

Stuff

Let

measure_theory_dc4cec3b8987e7361188a12c07a601150ff17f92.png

where measure_theory_13fd4a0ba05def0c160c0702c518237162764ea7.png are a set of positive values.

Then the integral measure_theory_eae4f89be054f458543db3ba1196a0d160895692.png of measure_theory_078b85cd3478400338e3a1ee425c2a468644be7e.png over measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png wrt. measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png is given by

measure_theory_95cd0c1ac899736e54417a02f12e63b321a3d837.png

Let measure_theory_1612fe43ef853a2849dc6aeb65521c0ca93220dd.png be a nonnegative measurable function.

We define the integral measure_theory_146bb5c18c9d8aef943d1eebf2863f0bd37bcafa.png of measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png over measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png wrt. measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png by

measure_theory_517c98ac36d132e13d7280f54c6b0d8910a5833e.png

Let measure_theory_77a1f0542406d2c0394b5a579aadb978f4050ea8.png be a sequence of nonnegative measurable functions on measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png. Assume that

  • measure_theory_8c760836a89d9571a4976dd5d23e56aa2753857c.png for each measure_theory_4c91032750887d8dca229b906727af58e9268232.png
  • measure_theory_6d5de7306c52390e023a64f698a567c2842ab72c.png for each measure_theory_4c91032750887d8dca229b906727af58e9268232.png.

Then, we write measure_theory_30e952c59b33bc5bbeaa4a6ce661fc47e9603b9e.png pointwise.

Then measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png is measurable, and

measure_theory_d061b537445951e5c7185c57814abf4765a61ef7.png

Let measure_theory_a18c72d5ceaf0450287ad89e33a2bb049b376dc3.png. By Proposition proposition:limit-of-measurable-functions-is-measurable, measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png is measurable.

Since each measure_theory_617b87515c58d75e4faeff2a8dbc4b795e512b7c.png satisfies measure_theory_9980d70503ea986aa0c4231b694a91f064985853.png, we know measure_theory_4222f368466ae706b64a646c87f57c6d5b7c2e29.png.

  • If measure_theory_e05ddbadca334364f916ca6f127d4e24977ece0d.png, then since measure_theory_e40092c5f498e5c3c64a612ad2203c9060d84964.png and for all measure_theory_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png we have measure_theory_99e8f03ee873a88b258579068cde8cea49b95c2e.png, and measure_theory_01b6ecc75c8d131dbc0cf8e6d254a43de06f0411.png.

Let measure_theory_7bffb2f056d330e577b20ca3c288fa26fde13046.png and measure_theory_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png.

Step 1: Approximate measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png by a simple function.

Let measure_theory_5dc59e8e67581e3469d6fd42edc34acb3f86c387.png be a simple function such that measure_theory_6d81deee9c189a39a5be9d6e75dce968c143b50c.png and measure_theory_9be6af4cc72e6082bae54156ee2478a9d0753224.png. Such an measure_theory_5dc59e8e67581e3469d6fd42edc34acb3f86c387.png exists by definition of Lebesgue integral. Thus, there are measure_theory_f7b9ba07ad514cc2867fd8b2764427e9637f4768.png such that measure_theory_7be633cbcb1042642148843d566269644a2fd705.png, and disjoint mesurable sets measure_theory_c1431fdfa5c96e6edc4f4689c9c21fa65bea7feb.png such that

measure_theory_d698e86445862f4fbf3ba4672df9fdcbd84d745a.png

If any measure_theory_52c026f8c4176f2f5a6078cf5f320adb1e5c8003.png, it doesn't contribute to the integral, so we may ignore it and assume that there are no such sets.

Step 2: Find sets of large measure where the convergence is controlled.

Note that for all measure_theory_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png we have

measure_theory_9d1a1bf72d8c11bd1acfb544b5e1265099f08be1.png

That is, for each measure_theory_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png and measure_theory_4024f69021bbc9ac2133f1d6c6dc5e0c72c204e5.png,

measure_theory_7ebbb211401bde02be45aab3a1fc8230bf67bb5e.png

For measure_theory_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png and measure_theory_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png, let

measure_theory_ad2a38a18fd3e8046731af037d9c72590a21d30c.png

And since it's easier to work with disjoint sets,

measure_theory_63030b20c757bfeabca3fac11271a2676bb1e73e.png

Observe that,

measure_theory_de325df319a7a42eb7d88ae2c989fa359ea2c553.png

Then,

measure_theory_fcb7d019b3b86ff67e137c91e1432831395825ce.png

We don't have a "rate of convergence" on measure_theory_ea55dcf242875c2e70c36acf772034de2bf3a4a9.png, but on measure_theory_c1f33a74c03512059aa2fa729eecef5a92cc5c7e.png we know that we are measure_theory_42a7f277a88bc1e37cbbf95697565b3c99fd666e.png close, and so we can "control" the convergence.

Step 3: Approximate measure_theory_617b87515c58d75e4faeff2a8dbc4b795e512b7c.png from below.

For each measure_theory_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png if measure_theory_bcc768dae0932d614f16a2136d6e673347157ecb.png, then let measure_theory_a7daec60bcf9b28065e6ccac6dea3d28723a4835.png be such that

measure_theory_2b623c89f16e6f0c2585e8945415cae047c657a8.png

and otherwise, let measure_theory_a7daec60bcf9b28065e6ccac6dea3d28723a4835.png be such that

measure_theory_0637e1a5f394d3d2aa7d4b9717c6f583202e8808.png

Let measure_theory_84d9a8929cb2bbe2c9c35802f9c2de4f06acfb87.png, and let measure_theory_0b88162a31c0efb8ef2f96787e370b5ecc81e574.png.

For each measure_theory_7ffaf88b7f2a236a036cc23c7916f304283a145f.png, measure_theory_88893f4405f6c1e732e9aca1d5c99079b79d1f47.png and measure_theory_c6bc90de8df9bd520d221b957e9552cce9eb8dec.png we have

measure_theory_0a70362cda66d1199d1ff6cb173f540d52c5f3d1.png

Thus, measure_theory_cb392230692195be1ba8873143b268fa1a210278.png, and measure_theory_7c039f3ab029857f0a719d4833281d1a5a466fc1.png,

measure_theory_c9dc1f06e5c4f6821590e6edb72a6e56860ff81e.png

If there is a measure_theory_d8797d487554fc78e460f5843167fcb01b53e76b.png such that measure_theory_410846943f49ab8786739a270cc8d0a77d800c2e.png, then

measure_theory_47ef6499a8242303275b7ef978472be76d1e317c.png

Otherwise (if the integral is finite), then

measure_theory_808cb38b9fb3d347d0ff1719905965ce9405d7a0.png

For every measure_theory_6394f37864b8d19d6f80bde5ac02c1a3f3bd96f9.png and measure_theory_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png, there is an measure_theory_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png such that

measure_theory_f6330040521f17ce9e95947025d921342d6486b1.png

For every measure_theory_6394f37864b8d19d6f80bde5ac02c1a3f3bd96f9.png such that

measure_theory_058fa2da19aa990e89eb891e82cf3823b53f5cac.png

Therefore

measure_theory_59e6229e81ce820516938411690f7f08d3946b7c.png

Thus,

measure_theory_1ea2f0698b178726d7a7b04bc8c2834c1556585a.png

as wanted.

Let measure_theory_23726c31cc6c1501e8a079c2ca7b4e5dfde88913.png be any nonnegative measurable functions on measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

Then

measure_theory_2aeabe6a352a89ff66618728de22a220b7a554ec.png

Let measure_theory_cf2d42a72347a665d931d4ba00a3f4e79196d4f3.png and observe measure_theory_b91045bad23a5f716d792771c9bde010ec711e58.png are pointwise increasing

measure_theory_8a1278ff85b77458243d3c05e325798dcf35c062.png

Properties of integrals

Let measure_theory_b88799d5ec72ed146276eedc13cdcd9204986c5c.png be a measure space.

If measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png is a nonnegative measurable function, then there is an increasing sequence of simple functions measure_theory_23726c31cc6c1501e8a079c2ca7b4e5dfde88913.png such that

measure_theory_78a8c257dff5d988c546acaa581024989e30edfb.png

Given measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png as above and measure_theory_84feeb83da8919c3f995550cdfb7e928ee8b390d.png for measure_theory_3f10c4c7bfe94aa15c1f70a64a85bbadc543ad2b.png, let

measure_theory_fc30437ff5f8a7f39529630e24137441fa0267a1.png

and

measure_theory_8d6a9ecdfdf37b5dfe4423600460e0ed18620569.png

Or a bit more explicit (and maybe a bit clearer),

measure_theory_d4e39303cc3382ab862cc4d950be17dd3f775eed.png

For each measure_theory_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png, measure_theory_a3c371a6b83d4434d729646aa0aeded310884db2.png is a cover of measure_theory_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png. On each measure_theory_d86873ff6d09df9945592718671d46b31d713676.png we have measure_theory_9980d70503ea986aa0c4231b694a91f064985853.png, hence measure_theory_9980d70503ea986aa0c4231b694a91f064985853.png on entirety of measure_theory_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png.

Consider measure_theory_5ae6b36dba66b3d59a4a16a87b3f05f8bdab18de.png. If measure_theory_806212c04256a11b2f89ad23935226abcf26ad11.png, then for measure_theory_e17754593c18e45553b88d8ae02416f442c364af.png which in turn implies

measure_theory_50abf065862353538d03a96d0a0edc4fcf173bda.png

Hence measure_theory_f1ea051a8b8580668c2e5ed4a2f16c8e5fc58ae4.png.

Finally, if measure_theory_d69c7d41ec3e094108f293db398a5d333ac3796e.png, then measure_theory_801417597dca106c6a0eb373d32010a3f8524bd2.png and for all measure_theory_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png measure_theory_617b87515c58d75e4faeff2a8dbc4b795e512b7c.png take on values

measure_theory_1f21250edb3fdbb773d198e449c6454997f38653.png

Hence, measure_theory_86fff707df410a65631e9a7f612d32ddf7486c48.png for all cases.

Furthermore, for any measure_theory_84feeb83da8919c3f995550cdfb7e928ee8b390d.png and measure_theory_c3590d46aa23b2ca67c99045e1fda9fb4bd99abb.png, there is the nesting property

measure_theory_f73825aa42159f460b9934a1d412dc32ca0760ff.png

so on measure_theory_d86873ff6d09df9945592718671d46b31d713676.png we have measure_theory_5cfce0f383bc78579e2e6bce03f1fbb05bc79fe6.png.

(This can be seen by observing that what we're really doing here is dividing the values measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png takes on into a grid, and observing that if we're in measure_theory_d86873ff6d09df9945592718671d46b31d713676.png then we're either in measure_theory_4b5b6a2758e1ad74a65c58d96129d2e6aa58e4a2.png or measure_theory_33a2d769bcef05ceda56b8855434b2dafe83fa43.png).

For measure_theory_47c9b1cca2c3049f00c29cf01abb42e3afb2f875.png, then

measure_theory_f5106bc910f6c91a8c5c650609628c8d3c189bbd.png

so again measure_theory_1833d09be2979e1b6c119609fc196e8826cb6d18.png and measure_theory_23726c31cc6c1501e8a079c2ca7b4e5dfde88913.png is pointwise increasing.

Let measure_theory_b88799d5ec72ed146276eedc13cdcd9204986c5c.png be a measure space.

Let

  • measure_theory_198a64800b20de12f4aafa400d8ff0f00fd78fb2.png be nonnegative, measurable functions
  • measure_theory_06766bbcfbc24a7658b188ad69035fc95f3a26bf.png s.t.

    measure_theory_3a77731c4caeccdd710e2872d16ee691c14ceb0d.png

    is defined

  • measure_theory_2ab47814406a32afc5d6db23bca34f6c335907e2.png be a sequence of nonnegative measurable functions.

Then

  1. Finite sum

    measure_theory_adeecd26a2a1d5ff8c30f7941a3ef69ab5fb306b.png

  2. Scalar multiplication

    measure_theory_87a1a2839a7119be99b1699be2c7885f2ed84724.png

  3. Infinte sums

    measure_theory_bc9124026c29f100146adaf576b39476b7e2fefe.png

Let measure_theory_23726c31cc6c1501e8a079c2ca7b4e5dfde88913.png and measure_theory_b91045bad23a5f716d792771c9bde010ec711e58.png be increasing sequence of simple functions converging to measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png, measure_theory_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png, respectively.

Note measure_theory_1f179799a7a0d439e6e551a76a95b33557da9b98.png is aslo increasing to measure_theory_7462dd536b01138161b93884a77eaecae30da461.png.

By monotone convergence theorem

measure_theory_b59279baf2be94734dd262a944af8474f8a40cd7.png

The argument is similar for products.

Finally, measure_theory_db79df93f1098a50b60f8b7322186d1a9e8637c8.png is an increasing sequence of nonnegative measurable functions, since sums of measurable functions is a measurable function.

Thus, by monotone convergence and the result for finite sums

measure_theory_aa7324e0b188ee1ec86bdef60c3c92d2d4862662.png

Integrals on sets

Let measure_theory_b88799d5ec72ed146276eedc13cdcd9204986c5c.png be a measure or probability space.

If measure_theory_df74da05ed7e51e806037dcdc01e51379d2f5176.png is a sequence of disjoint measurable sets then

measure_theory_baaf5552b89838af952a8e2dc8532563f8f8e010.png

Let measure_theory_b88799d5ec72ed146276eedc13cdcd9204986c5c.png be a measure or probability space.

If measure_theory_5dc59e8e67581e3469d6fd42edc34acb3f86c387.png is a simple function and measure_theory_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is a measurable set, then measure_theory_3fe53d4e51e33d2e1f0b6d35a10e02071d3ca91f.png is a simple function.

Let measure_theory_b88799d5ec72ed146276eedc13cdcd9204986c5c.png be a measure or probability space.

Let measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png be a nonnegative measurable function and measure_theory_41e6ce2ab80715fe484d74efc80263e1528225f7.png.

The integral of measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png on measure_theory_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is defined to be

measure_theory_851e9c7cf3540973353ba93a9f459cf1bc5623bc.png

Let measure_theory_b88799d5ec72ed146276eedc13cdcd9204986c5c.png be a measure or probability space.

Let measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png be a nonnegative measurable function.

  • If measure_theory_d628cc96f9875171d7e1e6e93e1912621b319d2b.png and measure_theory_79463e3ac84fc19e2ffdf1cc595c89299b9c5ea1.png are disjoint measurable sets, then

    measure_theory_0daddce2b95dcf43b6252c9844bcadc723598e73.png

  • If measure_theory_df74da05ed7e51e806037dcdc01e51379d2f5176.png are disjoint measurable sets, then

    measure_theory_57fd2f8d0f03ab72831c80824c9656d543854561.png

Let measure_theory_b88799d5ec72ed146276eedc13cdcd9204986c5c.png be a measure or probability space.

If measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png is a nonnegative measurable function, then measure_theory_381c79c765a0a25c255c0970a572b5bdc7e50ab7.png defined by measure_theory_3d1eeb02c2154ed59cd52f2b010caaf901245c49.png:

measure_theory_523f47bae8aca132e984684299b22fe1b0c38bf4.png

is a measure on measure_theory_37d3d2a40172b2cf7b0e8ea91ec033c3e1f9ecbd.png.

If measure_theory_af0d07640b2b34b6d26325db2db3cd8e64aa5ee9.png, then measure_theory_9f6cf0003fa4787ea36ff5edecec29b019be2fb7.png defined by measure_theory_3d1eeb02c2154ed59cd52f2b010caaf901245c49.png:

measure_theory_b3d252bc29c34952c00827d647c8a29c36cd9939.png

The (real) Gaussian measure on measure_theory_492d525117d0dcc93d066c8759f46b98cf9980ca.png is defined as:

measure_theory_6d53f44a1463706ea8ed9ecee7d34f7902e0c5cb.png

where measure_theory_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png denotes the Lebesgue measure.

A Gaussian probability measure can also be defined for an arbitrary Banach space measure_theory_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png as follows:

Then, we say measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png is a Gaussian probability measure on measure_theory_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png if and only if measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png is a Borel measure, i.e.

measure_theory_640ae35bff1732f65516cbc71f018d5e2cb12b33.png

such that measure_theory_3646403810da62cdd89fa03eebac80852c1eb884.png is a real Gaussian probability measure on measure_theory_492d525117d0dcc93d066c8759f46b98cf9980ca.png for every linear functional measure_theory_26d15ce421e8cb5a467df3aaab5d079d4c589c1b.png, i.e. measure_theory_37e1c29d8f1f869082402674d611a4d6b7c5dc32.png.

Here we have used the notation measure_theory_86088d5789b045a5569911751979c8e413984ba4.png, defined

measure_theory_467533a6d76abf1f8dedd169e92e3fff24e76907.png

where measure_theory_35a5cde93281f6aebef070b22cb194d3c970cece.png denotes the Borel measures on measure_theory_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png.

Integrals of general functions

Let measure_theory_b88799d5ec72ed146276eedc13cdcd9204986c5c.png be a measure or probability space.

If measure_theory_471070b4dd8fd5348430460c6ee2f9983fd7f933.png is a measurable function, then the positive and negative parts are defined by

measure_theory_3b856bdeae8a5d42035c93afc2b85eeb2b43d3c5.png

Note: measure_theory_a32b25a1c96fddac7682a49e94eef90fb00f9846.png and measure_theory_448d51c32b11750fef2bb1b9af0c568c1ec4f9c5.png are nonnegative.

Let measure_theory_b88799d5ec72ed146276eedc13cdcd9204986c5c.png be a measure or probability space.

If measure_theory_471070b4dd8fd5348430460c6ee2f9983fd7f933.png is a measurable function, then measure_theory_a32b25a1c96fddac7682a49e94eef90fb00f9846.png and measure_theory_888fde972f204692c6231c55b50cb1ee165a69be.png are measurable functions.

Let measure_theory_b88799d5ec72ed146276eedc13cdcd9204986c5c.png be a measure or probability space.

  • A nonnegative function is defined to be integrable if it is measurable and measure_theory_fd7943f8fe34cfd731dad126ae24ae88d523315a.png.
  • A function measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png is defined to be integrable if it is measurable and measure_theory_db4d9f7a378b061eda52597e00fb91a75113d8be.png is integrable.

For an integrable function measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png, the integral of measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png is defined to be

measure_theory_5cbb0378dba6f5e7781d4c657a3d9892de4a1215.png

On a set measure_theory_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png, the integral is defined to be

measure_theory_121d1fb5509f4f0137c934f35c74fccb3b6f324e.png

Note that measure_theory_db4d9f7a378b061eda52597e00fb91a75113d8be.png, but in the actual definition of the integral, we use measure_theory_42c41022d7e6fa1d18e6ee1baf9bc9c0007719c6.png.

Let measure_theory_b88799d5ec72ed146276eedc13cdcd9204986c5c.png be a measure or probability space.

If measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png and measure_theory_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png are real-valued integrable functions and measure_theory_f0abf6ff32d5d2aef08884d47da0c496d0559dc7.png, then

  1. (Scalar multiplication)

    measure_theory_5289f5089842b5a116b758678bae3ffc3bec1d3d.png

  2. (Additive)

    measure_theory_791aab6e82fd0d2d4a42d23e016b6cffc74f707e.png

Let measure_theory_b88799d5ec72ed146276eedc13cdcd9204986c5c.png be a measure or probability space.

Let measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png and measure_theory_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png be measurable functions s.t.

measure_theory_f20c651b6119909e8d8ba67946cd1fdf1884d9fb.png

If measure_theory_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png is integrable then measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png is integrable.

Examples

Consider measure_theory_492d525117d0dcc93d066c8759f46b98cf9980ca.png with Lebesgue measure. Is measure_theory_0180308bece4d2f176ceb2e4b35d0b3eacea6d4d.png integrable?

measure_theory_c82cbe8049ee9adb2b4b21aaa252eeebf72da0f3.png

And

measure_theory_0e87949575cb4351ada3bd9f026a82799c160214.png

and

measure_theory_c787aea3d2b60394c2eb2502096a1667df6abfe0.png

therefore

measure_theory_3813376299ce2f4d9252c738a43c5963e0963847.png

Thus, measure_theory_0180308bece4d2f176ceb2e4b35d0b3eacea6d4d.png is integrable.

Lebesge dominated convergence theorem

Let measure_theory_b88799d5ec72ed146276eedc13cdcd9204986c5c.png be a measure or probability space.

Let measure_theory_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png be a nonnegative integrable function and let measure_theory_076d048799f38dd9c725419f7b2b76da23cdbb38.png be a sequence of (not necessarily nonnegative!) measurable functions.

Asssume measure_theory_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png and all measure_theory_617b87515c58d75e4faeff2a8dbc4b795e512b7c.png are real-valued.

If measure_theory_cb392230692195be1ba8873143b268fa1a210278.png and measure_theory_84feeb83da8919c3f995550cdfb7e928ee8b390d.png such that

measure_theory_23833eebf9f089d6b1e6fa518e179db22c5c9c0c.png

and the pointwise limit

measure_theory_91a3b979a9904c10a294e53389487363466f11e3.png

exists.

Then

measure_theory_41c502ac63058b57845a6c0153012a9c47b91f22.png

That is, if there exists a "dominating function" measure_theory_fc41b1d76d936bd6313efbd911e604cdba009535.png, then we can "move" the limit into the integral.

Since measure_theory_0d6385680f9f50537637e277cd21d9caef3c8836.png and measure_theory_5ae6b36dba66b3d59a4a16a87b3f05f8bdab18de.png such that measure_theory_5ea8dd2b0a390ed5951e03c689c39a3965df603b.png, we find measure_theory_0d6385680f9f50537637e277cd21d9caef3c8836.png that measure_theory_afb2ea1a18317ba1946318fa828a21e0a90ee3c4.png and measure_theory_e4fa56590ffcc590cbc5a60599c2d1a66ce6705a.png are nonnegative.

Consider measure_theory_6cb9d4f780a125011f134959fdc4ca13eadc1ebc.png

From Fatou's lemma, we have

measure_theory_0bda11c7592a0bee99c40a412f999d5125e770d9.png

Therefore

measure_theory_75141eaa416714e9047f5c82f799b1f2a8fccf34.png

Consider measure_theory_32666edbad48091451ef6e1ce3cfcde38669a2c7.png, then

measure_theory_3d540d9404c475eedda85ee303a565d6e6d93db4.png

(this looks very much like Fatou's lemma, but it ain't; measure_theory_617b87515c58d75e4faeff2a8dbc4b795e512b7c.png does not necessarily have to be nonnegative as in Fatou's lemma)

Consider measure_theory_d7d58d1d44b120e70b4be72832f300f49829a71e.png

measure_theory_d9d234a19a16a0f249cc7d4f314fb4b4302d040b.png

Therefore,

measure_theory_02f0aba5a26fd364e7861095f64d60b1d6aa89e4.png

Which implies

measure_theory_2e1a35cb816ce06ed4188631fef3e08fd31799f7.png

Since measure_theory_fb373a643c4b806e87abb4b7eb3084253b6764b0.png, we then have measure_theory_77053a4b3146295d02c537e38616b96a0791cd48.png exists and is equal to measure_theory_02c87e72bda36682c410b9883c069124f048258b.png.

Examples of failure of dominated convergence

Where dominated convergence does not work

On measure_theory_492d525117d0dcc93d066c8759f46b98cf9980ca.png with Lebesgue measure, consider

measure_theory_6a8a77ed50a9bc0b124d1ca2f018ffc6f8f43bb6.png

such that measure_theory_ba9072d87393e3812a3843f624f0900c7dde74cb.png instead of measure_theory_f7e0d1d18c9435c764f8c85958499236e3b3fca8.png as "usual" with measure_theory_b30f18902447c086ce8f421b2259326bdb3eb1e9.png.

Both of these are nonnegative sequences that converge to measure_theory_96f53e8f2667720f54bd85623f46cbf545733989.png pointwise.

Notice there is no integrable dominating function for either of these sequences:

  • measure_theory_617b87515c58d75e4faeff2a8dbc4b795e512b7c.png would require a dominating function to have infinite integral, therefore no dominating integrable function exists.
  • measure_theory_8dd4aeab389d39dfc23c0fc2d9d5a5a7a9fd5432.png on the right, and so a dominating function would have to be above measure_theory_9e283f0a770977b9c5776238e85e4b1f2406cdd4.png on some interval measure_theory_563d650d19dae3376adf923c759a97164f4e6f77.png which would lead to infinite integral.

Thus, Lebesgue dominated convergence does not apply

measure_theory_dddb2b9970910c3394ef1390568c5c0d8779c528.png

Noncummtative limits: simple case

measure_theory_c975dde8a30d049ee29e59913dac7a90478c8910.png

Noncommutative limits: another one

Consider measure_theory_492d525117d0dcc93d066c8759f46b98cf9980ca.png with Lebesgue measure and

measure_theory_dd6c545aba0336d15e9eb0461ff927e800586707.png

Consider measure_theory_2de3afd9599237e3ae9eedf5cc28ccb81e103612.png and $ b > 1$ and

measure_theory_ae41460cbe797df18c89efe11f46b93a0e83d383.png

Note that measure_theory_dcc52db96ce3f36bd936e6a4d23a5b1a44000f62.png, so measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png is not integrable.

Consider

measure_theory_759f65d3fac4e35d735495a89134bbe5bd420a27.png

Commutative limits

Consider

measure_theory_071a5953314d2f8ef51f57a8e4a62989654bfb06.png

measure_theory_facacfa441963fd12cb3925c56028e705e174744.png

We know that measure_theory_c1ecbd6ec3e2896e17790d0ef82c1cb3cfc66a4b.png is integrable and for all measure_theory_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png and measure_theory_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png,

measure_theory_4371eff7be4f40518a6831502dea5e415ba9cc90.png

By multiple applications of LDCT

measure_theory_3c25937c92ab78d41319bc7ec7dc4ab8696f164a.png

Showing that in this case the limits do in fact commute.

Riemann integrable functions are measurable

All Riemann integrable functions are measurable.

For any Riemann integrable function, the Riemann integral and the Lebesgue integral are equal.

Almost everywhere and Lp spaces

If measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png is a nonnegative, measurable function, and measure_theory_bc91ffd8a4c75bd2eca3c0ede260abea348c15fe.png, then measure_theory_ec4ed0dc3c9808c4d88815da3e14b61fda283d86.png.

For measure_theory_fe2c53a21d8a6ea03f4395abf662960ba035999b.png, let

measure_theory_81e85c561589e9377f81818b1c0d95e7da875a31.png

Observe the measure_theory_2c664f9cd54307d502a010b32e8e9b5d724ed2eb.png are disjoint and

measure_theory_eb27f5b455a2ec0bcc45463266091354e91cc2fc.png

Suppose that measure_theory_f7cc0619ca4898cfcf54032a30ac0e5b2139e825.png. This implies that measure_theory_1aa7867117653bad08f91b90db9f117bf9609a82.png on a set of positive measure, i.e.

measure_theory_5e0eb1309248616a14062fefe2649ba93d9abf01.png

but this implies that

measure_theory_8c0a55b6e0e913112cacf87be4d06488522912b2.png

Thus,

measure_theory_5085b8ae2059537b97003b68c84564fbd65ec7bd.png

which is a contradiction, hence measure_theory_ec4ed0dc3c9808c4d88815da3e14b61fda283d86.png.

Let measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png and measure_theory_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png be integrable.

measure_theory_6d4ba5efa91183eeb350439b9a5e555e998c519f.png

measure_theory_5ad51f7fa103dde0a0c76226b7f41f2f7eec4f34.png is the set of all equivalence classes of integrable functions wrt. the equivalence relation given by a.e. equality, i.e.

measure_theory_b359eb0142df05fcbaa5a6923c7ba8e4def10d4c.png

If measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png is an integrable function, the measure_theory_b6c157bdb77e1a2a2b8fc979c90b4731ca97c123.png norm is

measure_theory_84ea36100fac7828663a638add803d87f58c92c5.png

If measure_theory_3257c00004ee70d26408214acec67821ab80bf4e.png and measure_theory_1a36591465874fdc8ac9f5cf94a76fb918c63333.png, the integral and norm are defined to be

measure_theory_cf477e453acb5cbecb241c9a6ec17a7c529ae970.png

If measure_theory_fd8ed584985833d21065950c41fd7947d1894d47.png, then measure_theory_77d198ec7b058b0d0e24bc125b794475fe16e8e4.png, and

measure_theory_a4316a9827bebfa9b2b1801d94ab54cabbe1d8da.png

measure_theory_b6c157bdb77e1a2a2b8fc979c90b4731ca97c123.png is a real vector space with addition and scalar multiplication given pointwise almost everywhere.

Functions taking on measure_theory_0d14c6438211770a2ec19231c594e05f6320a51e.png on a set of zero measure are fine!

These functions are still the almost everywhere equal to some integrable function (even those these infinite-valued functions are integrable), hence these are in measure_theory_b6c157bdb77e1a2a2b8fc979c90b4731ca97c123.png.

measure_theory_b6c157bdb77e1a2a2b8fc979c90b4731ca97c123.png with the metric

measure_theory_cf8c69265f5e07dea7e3441091620916f80bf394.png

is a complete metric space.

Let measure_theory_8bb0798cea41d7eba131d401bab88417b18ee1f5.png be a Cauchy sequence. Since the measure_theory_2d8451421f69781e044825756e911af8aa0ab2f2.png are integrable, we may assume we choose measure_theory_492d525117d0dcc93d066c8759f46b98cf9980ca.png valued representatives.

For measure_theory_4028f9016d4f7a233195bc931939fa3360d350cd.png, let measure_theory_b33fef33e808887160154a9f5113efd7b0c559c9.png be such that for measure_theory_37d8c8c822e4bf340c41c41c67f465d938208354.png,

measure_theory_4dfad9788c6a1b310ad1397befe814d1644558c1.png

and measure_theory_4f6cc593298c0d20a96d760fe23e057698136142.png.

Thus,

measure_theory_c0e8f80870b6fa3cb08c5b5cf6d4771c92eb381f.png

and

measure_theory_e88b79375e2e469f19b2d12277026535622fc54a.png

Thus, measure_theory_d7b92d0136415f5c9aa3aa0b48cbec651147914c.png is finite almost everywhere. Thus, this series is infinite on a set of measure zero, so we may assume the representatives measure_theory_2d8451421f69781e044825756e911af8aa0ab2f2.png are zero there and the sum is finite at each measure_theory_5ae6b36dba66b3d59a4a16a87b3f05f8bdab18de.png.

Thus, measure_theory_58cb1509d4c716b8597b5626ba226a4abde28506.png converges everywhere.

Let

measure_theory_9d3f31d5418eb092c92d1187a22519357400d6b2.png

(observe that the last part is just rewriting the measure_theory_abd2e429d06e71e61489753cc022c110e23462f6.png).

By monotone convergence theorem

measure_theory_3208285d0aa3999d3ec0fe32cd18fece7dbd9016.png

Observe that pointwise

Applications to Probability

Notation

Probability and cumulative distributions

An elementary event is an element of measure_theory_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png.

A random event is an element of measure_theory_37d3d2a40172b2cf7b0e8ea91ec033c3e1f9ecbd.png

A random variable is a measurable function from measure_theory_5537083cdab42c7108d37538c78dc872ac285109.png to measure_theory_6a76f3a5fc1cad4fa88fbc8b6bf7c407338d2a3c.png.

The probability distribution measure of measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, denoted measure_theory_a58592e1d234811fe21b9434948cc8d9c92dbd0c.png, is defined

measure_theory_99fe1bca327819224fe0099eb75133c4d4a52478.png

The cumulative distribution function of measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, denoted measure_theory_9ca8d4a3e458c0a1c16591dc9c28e40f5e517841.png, is defined by

measure_theory_fe756dea914667d822d843c86da5ac4148f8fdd6.png

The probability distribution measure measure_theory_7213812d302e3f5c0984bfd3448e3e81d9240b23.png is a probability measure on the Borel sets measure_theory_57374af10bfdcfe66f9ad361e98cbc9c879f5ec7.png.

measure_theory_98a4f430e8d6584c1f41bac3a255d412da0142e6.png

If measure_theory_100791d2186e70bd7a69bf6f202092e8c5566932.png is a disjoint sequence of sets in measure_theory_f82f3967df7ce419929feecb902cf0d722024d10.png, then

measure_theory_b99751a2eb62ce320dc1dfebfdf2d1db1be34942.png

so measure_theory_7213812d302e3f5c0984bfd3448e3e81d9240b23.png satisfies countable additivity and is a measure.

Finally,

measure_theory_433d8c284744db83ecfd3ebf72613a336a5895c6.png

so measure_theory_7213812d302e3f5c0984bfd3448e3e81d9240b23.png is a probability measure.

  1. measure_theory_decad2ad3dd314b86a469e92f7b21c0f8d91fef1.png is increasing
  2. measure_theory_52288fb3d80db6a1511a443547c16124f0c6d32a.png and measure_theory_04411aa6f5e91cb63ce4668f7829ca28d4ef576c.png
  3. measure_theory_decad2ad3dd314b86a469e92f7b21c0f8d91fef1.png is right continuous (i.e. continuous from the right)

    measure_theory_106242b72044381dfe0b2227a791aba9d6eefbcf.png

  1. If measure_theory_7761d72b6c268e6de08424386c053652abb991dd.png, then

    measure_theory_a2d32493432b7a84a969bbe6defd44e595654aa1.png

  2. Consider the limit as measure_theory_407caa29cd42f9b2436f8953e3a8766db638e038.png. Let

    measure_theory_1486fc3a1880ec6cd9942a0327848dd9500470a3.png

    so

    measure_theory_6ddf94ef80b732727601aae661eceb0519d23c6d.png

    Then,

    measure_theory_6520937c3843e1cf01d4dcc546c9c28de372401b.png

    which, since measure_theory_decad2ad3dd314b86a469e92f7b21c0f8d91fef1.png is increasing implies

    measure_theory_50957a799d0e5c12953c61f70c45d3142b441b50.png

  3. Let measure_theory_b0706adb445a2c20a4d236cf389c6138cf2c5ccb.png and measure_theory_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png. Let

    measure_theory_f10e6aa87d667630b3d2a4b6bb7eed60852bce97.png

    The measure_theory_488c5b0b15f78befa3f04969b097b5afe3c8061e.png are nested, and similarily measure_theory_01c0eed0c94c8a74faf5db8c106e044ac0d65b99.png are nested.

    measure_theory_f98a55277778387963bdd3f47d12988af684642e.png

    Thus, given measure_theory_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png, there exists measure_theory_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png such that

    measure_theory_6e5f55803e55011fe60df697ed5b457e6b256424.png

    Let measure_theory_62037dfb9f0c17f781b95accb84ef2131d79fab7.png so

    measure_theory_ae96e6b4430de703d2a193f1b22b2122b62a2a06.png

    measure_theory_b93599477ded30f9176e91137f308cb953e9be57.png

Radon-Nikodym derivatives and expectations

Let

The following are equivalent:

  1. measure_theory_068aca6e7397c7fcde21b468a28806989242c5dd.png
  2. measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png is a Radon-Nikodym derivative for measure_theory_7213812d302e3f5c0984bfd3448e3e81d9240b23.png wrt. measure_theory_5f4a25dcd0aa83d676621d9e17960383e3a36a32.png (the Lebesgue measure but restricted to Borel measurable sets)
  3. measure_theory_980d9f68d83229e02af8d79c4ebb3430e47405d2.png

(2) and (3) are immediately equivalent:

measure_theory_615d96381e108787b6a34c29a1546288ec794f6d.png

iff (2) or (3) holds when considering only sets of the form measure_theory_0e5dc3a457e704b97b39da2797d35fc57f0a60db.png.

This statement is also equivalent to (1).

Thus (1) is equivalent to (2) or (3) restricted to sets of the form measure_theory_66924ac4617eb2c7b96d66999be94f08337b48ca.png.

However, sets of the form measure_theory_66924ac4617eb2c7b96d66999be94f08337b48ca.png generate measure_theory_f82f3967df7ce419929feecb902cf0d722024d10.png, so from the Carathéodory extension theorem this gives measure_theory_9a39983c45cfb82fbec9a8840b2438dcb966a820.png.

To prove measure_theory_1b5ac127fc0e1154c246cf565195a13d5990439b.png more rigorously, let

measure_theory_3d567177f74408edf106c1ce31b2d776186d23a1.png

for measure_theory_c9df15b7a83a759192c0f70cc3635c7d8b977e51.png s.t. measure_theory_9bb982ee7e4de113336f2d732cb6cb7f6b5956f4.png and none of these intervals overlap. That is all finite unions of left-closed, right-open, disjoint intervals.

Also let

measure_theory_61f50d6706e1891d7ad111f51808f15259874a11.png

Observe that

measure_theory_373dd1cd21595fee916b413bc722a231203aa770.png

and that

measure_theory_b3bcef98eb82490dc114c000fd75d5b9cb912278.png

One can show that measure_theory_98858b051bf04071d99125fb261235b91edfce29.png is a premeasure space. Therefore, by the Carathéodory extension theorem, there is a measure measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png on measure_theory_57374af10bfdcfe66f9ad361e98cbc9c879f5ec7.png s.t.

measure_theory_a5fb61dddc00dd0bfce7b17bb05d250cf4968490.png

Furthermore, since measure_theory_f037415c0c4a5d6b4eb827ef5dbc3c123acc84a2.png, measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png is unique! But both the measures measure_theory_bfe20ce4e0c0e488166515c2a260282a6d83d2c5.png and measure_theory_43f13a701907be2cd3e694e301f4847482955061.png satisfy these properties, thus

measure_theory_a7947fd191463a1dc9d0d51c403e07d10b6ea157.png

which is the definition of measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png being a Radon-Nikodym derivative of measure_theory_7213812d302e3f5c0984bfd3448e3e81d9240b23.png wrt. Lebesgue measure restricted to the Borel σ-algebra, as wanted.

A function measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png is a probability density function for measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png if measure_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png is a Radon-Nikodym derivative of the probability distribution measure measure_theory_7213812d302e3f5c0984bfd3448e3e81d9240b23.png, wrt. Lebesgue measure restricted to Borel sets, i.e.

measure_theory_5c5c856e18f40af001eae962f1cd9cb2da4a5d11.png

Expectation via distributions

Expectation of a random variable is

measure_theory_c03cbdc3ba1b839663c1cbca26a41ddf2dfb47de.png

If measure_theory_73f39201c8fda74358962d12d3476e0251ed1300.png is a nonnegative function that is measure_theory_f82f3967df7ce419929feecb902cf0d722024d10.png measurable, then

measure_theory_6b160d12bbe6fa954588544199ef86e2043c1b84.png

If measure_theory_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png is the characterstic function, then, if measure_theory_e6e9bd287b92d383348cd65615015fdbe22644c5.png,

measure_theory_26e4b245500c41d3b8480624b4cd6b885adc2502.png

so

measure_theory_a0244dad8e66037c7af900659ff613583a7fe640.png

Multiplying by constants and summing over different characteristic functions, we get the result to be true for any simple function.

Given a nonnegative function measure_theory_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png, let measure_theory_db52622c260eabbd1db4af104c8cb92a9a899c73.png be an increasing sequence of simple functions converging pointwise to measure_theory_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png.

Note measure_theory_e0c210485222a3237cad14a7eba975225e6b4f9c.png is the increasing limit of measure_theory_189896d8b2d9b4487daf00af1b5e5902db7de310.png. By two applications of Monotone Convergence

measure_theory_57bb1258eeffec16c877335f25bb69f1075166eb.png

This techinque, of going from characterstic function → simple functions → general functions, is used heavily, not just in probability theory.

Independent events & Borel-Cantelli theorem

A collection of random events measure_theory_4bb678eec638f7df86e2f1d6282e6a5dc723b058.png are independent events if for every finite collection of distinct indeices measure_theory_2dae7da1c66cba533bbbef24d326606c201094a5.png,

measure_theory_5eb3b1d9e07cf076217659ee3776b057cad6bfab.png

A random event measure_theory_979227849a36a75ef8364c3869b9c4c1fe5cf97d.png occurs at measure_theory_5ae6b36dba66b3d59a4a16a87b3f05f8bdab18de.png if measure_theory_ac0b04fb01ef23618f50f8b62755ab7ab80d1d3c.png.

The probability that the event occurs is measure_theory_2f68c99fdc95411b73dbadc4d525669eba9a1594.png.

If measure_theory_5f619617e80726b703c3ddd269a8df3e715f022a.png are independent then measure_theory_08fbb140589a4d7b089ff44cd078c1d8fe9c2b78.png are also independent.

Prove that measure_theory_f86960f2b195e3fefa8f0d8df7339932ad6fdd80.png are independent.

Consider measure_theory_589d6d3cc1178d5422265156ece8469a52c7aebd.png, we want to prove

measure_theory_86d965bb570360cf02207eee155fcea78e50f32a.png

RHS can be written

measure_theory_b87d71d64dd786b7c793de95828025d9b5408cd0.png

which is equal to LHS above, and implies that the complement is indeed independent.

The condition that infinitively many of the events occurs at measure_theory_e29af05574032ace665d996d46b3280fc49866ef.png is

measure_theory_0aefbe8f87212d2f00bb1d0051b0c421a65a09a0.png

This is equivalent to

measure_theory_34bca995cd178790753609d81c81f0537a753b85.png

where we have converted the measure_theory_254a46746eef6b44bfd9131911fe826e4b87a248.png and measure_theory_f6766dd80e1fcb6df5c2f8790ee367125640ed0d.png.

Furthermore, measure_theory_932957fb985f6361c0e68d485d6d37e594d5288b.png is itself a random event.

  1. If measure_theory_e5636686c352f25a4c0c8e2ffd692a054c7294ff.png then probability of infinitely many of the events occuring is 0, i.e.

    measure_theory_a7767a5b1a1f76547075e70113f3760ae738ffe5.png

  2. If the measure_theory_d6d5a6d88bd930a55f2ff81079f4c752c6f5d933.png are independent and measure_theory_31e4d7a22ee314a3f1ef93b5ca0ba8abb0515c94.png, then probability of infinitely many of the events occuring is 1, i.e.

    measure_theory_11b114edcd53e11a11efa08a42d0749d0dbc6731.png

  1. Suppose measure_theory_17518ce7ac6e8d2471523ea6e0ad7e2b6cc29dc2.png.

    measure_theory_43ae21d0123c3b822f80cff937b4e9688595ccc3.png

  2. Suppose measure_theory_d6d5a6d88bd930a55f2ff81079f4c752c6f5d933.png are now independent and that measure_theory_31e4d7a22ee314a3f1ef93b5ca0ba8abb0515c94.png. Fix measure_theory_84feeb83da8919c3f995550cdfb7e928ee8b390d.png. Then

    measure_theory_db4be0c0ea3997e9559cbb09f11ccf9c7747e36f.png

Chebyshev's inequality

Let measure_theory_bb22d7e3135e825adc3652a3a51866d686beb4d3.png be a probability space.

If measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is a random variable with mean measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png and variance measure_theory_68668609fc72c879ba0fabd0c25275964a5e4af8.png, then

measure_theory_3a3e3e8f0a659ad6a716579bd010399c51abf668.png

Let

measure_theory_10e3e4bf0c362dc0755e85f10502b6340a43aa66.png

Then measure_theory_c79cfdfd35cb3618673026a7d3558e769f2a0df5.png everywhere, so

measure_theory_f9c74b39eb01d70e6c03e8a152354cc7c06a3478.png

Hence,

measure_theory_c86bad5722152ee36050e0acaa5380e0a2326ce3.png

Independent random variables

Let

  • measure_theory_bb22d7e3135e825adc3652a3a51866d686beb4d3.png be a probability space.
  • measure_theory_602479be5e5c6c6898e0b72acd59e63e0f6f6213.png

A collection of σ-algebras measure_theory_00b79b7ed51e30898a2a9c4ef450a3dd7a5d921c.png, where measure_theory_acd5374faa2d76b082dcc76bf69419e84b74b7ca.png for all measure_theory_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png, is independent if for every collection of events measure_theory_fbb1e2e08d28a3d6eb46f960d091e4da562f4af9.png s.t measure_theory_8e61a8a16dfba900e1c78ef4c2492775b1b8ec0a.png for all measure_theory_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png, then measure_theory_fbb1e2e08d28a3d6eb46f960d091e4da562f4af9.png is a set of independent events.

A collection of random variables measure_theory_c6a456e0ff4cde1d5dddefc2f1ea3409434713a5.png is independent if the collection of σ-algebras they generate is independent.

A sequence of random variables measure_theory_593e9d6b658be0e97d66a1bdca6e498996a50911.png is independent and identically distributed (i.i.d) if the yare independent variables and for measure_theory_c92373d2b03b4a43413cfd5e8385039ba98ecf38.png we have

measure_theory_779036b5dd1c31e74948355a453264be0bbc6fe0.png

where measure_theory_5c6731507cc630689cbe452436cab3918c67ab5c.png is the cumulative distribution function for measure_theory_5ea13e10d0ae85fcb7e47caa15b5618926a4fc70.png.

Let measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and measure_theory_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png be independent.

  1. We have
    1. If measure_theory_d1d5bee4ce1449c5f7e1723a0cd2268a1ab935c6.png or measure_theory_176b55dec8e6c7c2f6e83fadaef2647206ccc9f0.png then measure_theory_ba4da7ba4db8836cc29f51e088855124b966dd98.png
    2. If measure_theory_87ccfe0ec466ee2c11169e94aade5cd7623e4ec9.png and measure_theory_8bb8dc8eba8fc40ceafa7fb33963f2145efd9e67.png, then

      measure_theory_92375abbe12309fb318ba47bedfc6f5b703023c1.png

  2. Furthermore, if measure_theory_fc07a575757da22f192c9ea8f8cf7939943d8e19.png and measure_theory_f14eee8473ac595bb083b650bfa6aed3000a93eb.png, then

    measure_theory_0eaa15ec0b91a96328058d246b20cdc25aff0c0c.png

Consider

  • first nonnegative functions
  • subcase measure_theory_8645335377ac824d8a059163660c0b1d27fea5b2.png

Since measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is nonnegative

measure_theory_4fce19e475e95d812e717e2cfbb038c640b9804f.png

Thus, measure_theory_a7a6a549ce0f23ac6bc33b456f559d03cc86c7d7.png so measure_theory_8ced94b95c2888b41d73c5040f26560d426054fc.png.

Now consider the subcase where measure_theory_d7c867816426438987fd60a7c258933b7a2a6660.png and measure_theory_0c0af6de366c5f2bcf4ef3989811e91adb867324.png.

Let measure_theory_f3661396175635ca9d530eb804b612080154b5ae.png and measure_theory_bf025db540d5f0f78df3409a4fc990035ad5c2b9.png be the σ-algebras generated by measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and measure_theory_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png.

Observe that measure_theory_3ae3ea302dc8556e8105bf3c1e7da2758428f3a6.png and measure_theory_de813878bbeac81df99c9a2b6eba8918ba497296.png are measure spaces. Let measure_theory_593e9d6b658be0e97d66a1bdca6e498996a50911.png be an increasing sequence of simple functions that are measurable wrt. measure_theory_f3661396175635ca9d530eb804b612080154b5ae.png and similarily measure_theory_3516bfb64c8a5e47bffa477688a4fdf6fa319d3c.png simple increasing to measure_theory_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png and measure_theory_bf025db540d5f0f78df3409a4fc990035ad5c2b9.png measurable.

As simple functions, these can be written as

measure_theory_6eeb9adac40f8572b5afda2975dcb025323cd1c1.png

Then,

measure_theory_4b4b587c53a9f4ab398b310138b69e55b58391e4.png

Since measure_theory_34910a4905971901900fdcf8fb89a0288470ba71.png increases to measure_theory_edb24343cde4419be10cc1df9a3a452a45e381bb.png, by MCT

measure_theory_0eaa15ec0b91a96328058d246b20cdc25aff0c0c.png

Dividing into positive & negative parts & summing gives measure_theory_39740af1263c8e9d75965873d956f0c0c9bbcab5.png.

Strong Law of Large numbers

Notation

  • measure_theory_388b030c646fff1ff355bc9e4ab86267da56da53.png are i.i.d. random variables, and we will assume

    measure_theory_96eb01d952c68980882f04f5a94c97773b460d53.png

Stuff

Let measure_theory_bb22d7e3135e825adc3652a3a51866d686beb4d3.png be a probability space and measure_theory_388b030c646fff1ff355bc9e4ab86267da56da53.png be a sequence of i.i.d. random variables with

measure_theory_4927b0ebef199f994c25754b5021133644d621b1.png

Then the sequence of random variables measure_theory_eed21cb787b07d66120df9bb01d3a45ba171ec20.png converges almost surely to measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png, i.e.

measure_theory_cd060d4c3e11a1caf6d9022fe8d4a7c59e83fdce.png

This is equivalent to measure_theory_ef453b1c74569a11c7b99713b879b3975f785bb1.png occuring with probability 0, and this is the approach we will take.

First consider measure_theory_51ec5a718b7d23e4260aa28cb686dca75c181975.png.

For measure_theory_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png and measure_theory_4028f9016d4f7a233195bc931939fa3360d350cd.png, let

measure_theory_07190f3dfc460f3e46e36783da59f714e3b25edf.png

and

measure_theory_7c272d3d8b7771e72d16d746df4126dde04aab65.png

Since measure_theory_a222366d7226f09851be223705fbdb373b863be0.png are i.i.d. we have

measure_theory_104fd897ed5ca85e8d98317f9d2388614a81f13e.png

and since variance rescales quadratically,

measure_theory_b39c2b9e034af4f55c7a082910756d176f1d6297.png

Using Chebyshev's inequality

measure_theory_b7f367da8c52461ef41ff39796690b4e6d3d301b.png

Observe then that with measure_theory_407321209930c3166bccd28cc8d9d86bc34023b0.png, we have

measure_theory_558e0426de12faf99ab2ee55c49f07098266a32a.png

And so by Borel-Cantelli, since this is a sequence of independent random variables, we have

measure_theory_61d0f693432a8e41621676ecf773debba78ed16d.png

In particular, for any measure_theory_224ae917d74dc2133d4403064c971bf562d4db50.png, there are almost surely only finitely many measure_theory_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png with

measure_theory_11b0b781c58437bdbbf09e268020a5e7a3dcace1.png

Step: showing that we can do this for any measure_theory_224ae917d74dc2133d4403064c971bf562d4db50.png.

Consider measure_theory_83d1123b6b5b8eed94e9c4d0d47d835a91feb721.png. Observe that by countable subadditivity,

measure_theory_eac95359b5ccdbb9e8b8b51c2890e0c70ffbf32e.png

Now let measure_theory_d5326cbc12d6451e494984767e107a16792c6d09.png, which occurs almost surely from the above. For any measure_theory_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png, let

measure_theory_ed0c856a1ba68964851d06407389be8a9454007f.png

Since measure_theory_d5326cbc12d6451e494984767e107a16792c6d09.png, there are only finitely many measure_theory_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png s.t.

measure_theory_3337f13f470dfc8729d02b06f2f6eccbb5281351.png

as found earlier (the parenthesis are indeed different here, compared to before). Therefore

measure_theory_c19a600f2c961647fbdae765b78c0da289ad0e3f.png

measure_theory_d402086a34ea5a67dbbf6f855d57b4a438956cb7.png is arbitrary, so this is true for all measure_theory_224ae917d74dc2133d4403064c971bf562d4db50.png. Hence,

measure_theory_6b456368087c442a54337923b4f1a9beb7182fe4.png

This proves that there is a subsequential limit almost surely.

Step: subsequential limit to "sequential" limit. Given measure_theory_84feeb83da8919c3f995550cdfb7e928ee8b390d.png, let measure_theory_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png be such that measure_theory_b800112cb698123589c0c47584a7eeb956cd3049.png. Since measure_theory_5ea13e10d0ae85fcb7e47caa15b5618926a4fc70.png are nonnegative

measure_theory_8807d6043df6e2956dab61629aa6a3c698c9d96a.png

and therefore

measure_theory_5e5be520f4911971f932cc3201280d950632ea5c.png

and since measure_theory_b800112cb698123589c0c47584a7eeb956cd3049.png,

measure_theory_4785b62e6beb1cc1b056a2adc1171b96abf3458f.png

Since the first and the last expressions converge to measure_theory_acb0106122b7f90d9bd5639367a141a7e53d8327.png,t by the squeeze theorem we have

measure_theory_96914a3548c9e1bf0f442d9909b2b5bf94c7ed70.png

Step: Relaxing nonnegativity assumption on measure_theory_5ea13e10d0ae85fcb7e47caa15b5618926a4fc70.png.

Suppose measure_theory_5ea13e10d0ae85fcb7e47caa15b5618926a4fc70.png is not necessarily nonnegative. Since, by assumption, measure_theory_5ea13e10d0ae85fcb7e47caa15b5618926a4fc70.png has finite expectation, measure_theory_5ea13e10d0ae85fcb7e47caa15b5618926a4fc70.png is integrable. Therefore we know that the positive and negative parts of measure_theory_5ea13e10d0ae85fcb7e47caa15b5618926a4fc70.png, denoted measure_theory_4f75d69a80e865547251affb0565bb5b3c0141de.png, are also integrable. Therefore we can compute the expectations

measure_theory_80874a6aa96796b2cb836acd4dc4cb78e173e7a6.png

Similarily, we have that the variance of measure_theory_4f75d69a80e865547251affb0565bb5b3c0141de.png is finite, which allows us to the apply the result we found for measure_theory_aa07b3a8458adb2855b54064282b1f340d44fbd4.png being nonnegative to both measure_theory_f59905e3f3740a283ea412d0984ccb609bbb2b1b.png and measure_theory_946d4a7c3df21f8971229d256cb0d1fe6f29f004.png:

measure_theory_6d6a90cda71916650159fff31a2c3edc294e1c99.png

Let measure_theory_1a662234ae7519bc18fd554d7664ebc2344d652a.png be the set where the mean of the positive / negative part converges. Since

measure_theory_507fb760e3135d3cb9698e77e8e9bdff63e93eed.png

(since otherwise the limit would not converge almost surely). We then have

measure_theory_5aec91cb1c71c694d2634e9c7a90af5f3c30bf5d.png

Thus, almost surely, measure_theory_9fd7e52f0facde4d94dc226e5d5d53a89c7f248f.png, and on this we have convergence, so

measure_theory_6f9770cef5b6685897b1845ac2f867b568687ef2.png

Concluding our proof.

Ergodic Theory

Let measure_theory_61e5589f0f948fdfe5922eb14ba15d50dc67c567.png be a measure-preserving transformation on a measure space measure_theory_85836c3c18ee764fbdef801c3631f37f2456dcf7.png with measure_theory_d5e912600ab8891e4a4e9ce716a36aafd21c0f9a.png, i.e. it's a probability space.

Then measure_theory_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png is ergodic if for every measure_theory_0eee8e67cb0b4331b9a6c9f62836d10e5c186781.png we have

measure_theory_ef05cd277cb9692f4df1dd03297e1ba60bb6f147.png