[PDF] Math 639: Lecture 1 - Measure theory background





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Math 639: Lecture 1

Measure theory background

Bob Hough

January 24, 2017

Bob HoughMath 639: Lecture 1January 24, 2017 1 / 54

Probability spaces

Denition

Aprobability spaceis a measure space (

;F;Prob) with Prob a positive measure of mass 1. is called thesample space, and!2 are called outcomes.F, a-algebra, is called theevent space, andA2Fare called events.Bob HoughMath 639: Lecture 1January 24, 2017 2 / 54

Algebras of sets

Denition

A collection of setsSis asemialgebraifIfS;T2SthenS\T2SIfS2SthenScis the nite disjoint union of sets ofS.Example

The empty set together with those sets

(a1;b1] (ad;bd]Rd;1 aiAlgebras of sets

Denition

A collection of setsSis analgebraif it is closed under complements and intersections.Lemma IfSis a semialgebra, thenS, given by nite disjoint unions fromS, is an algebra.Denition

A-algebra of sets is an algebra which is closed under countable unions.Bob HoughMath 639: Lecture 1January 24, 2017 4 / 54

Borel-algebraDenition

Given a collection of subsetsA

, thegenerated-algebra(fAg) is the smallest-algebra containingfAg.Denition

In the case that

has a topologyTof open sets, theBorel-algebrais (T).Bob HoughMath 639: Lecture 1January 24, 2017 5 / 54

Borel-algebraDenition

The product of measure spaces (

i;Fi),i= 1;:::;nis the set 1::: nwith the-algebraF1:::Fn=(Sn i=1Fi).Exercise Letd1. With the usual topologies, the Borel-algebraBRdis equal to B R:::BR(dcopies).Bob HoughMath 639: Lecture 1January 24, 2017 6 / 54

Dynkin'sTheoremDenition

A-systemis a collectionPof sets closed under nite intersections. A -systemis a collectionLof sets satisfying the following

2LFor anyA;B2LsatisfyingAB,BnA2LIfA1A2:::is a sequence fromLandA=S1

i=1AithenA2L.Bob HoughMath 639: Lecture 1January 24, 2017 7 / 54

Dynkin'sTheoremLemma

LetLbe a-system which is closed under intersection. ThenLis a -algebra.Proof.

IfA2LthenAc=

nA2L.IfA;B2LthenA[B= (Ac\Bc)c2L.Thus, iffAig1i=1is a sequence inL, then for eachn,Sn i=1Ai2L, and henceS1 i=1Ai2L.Bob HoughMath 639: Lecture 1January 24, 2017 8 / 54

Dynkin'sTheoremTheorem (Dynkin'sTheorem)IfPLwithPa-system andLa-system then(P)L.Bob HoughMath 639: Lecture 1January 24, 2017 9 / 54

Dynkin'sTheoremProof.

Let`(P) be the smallest-system containingP. We show that`(P) is a-algebra.LetA2`(P) and deneLA=fB:A\B2`(P)g.We check thatLAis a-system. I

2LAsinceA2`(P)

IIfB;C2LAandBC, then

A\(BC) = (A\B)(A\C)2`(P).

IIfB1B2:::is a sequence fromLAwithB=S1

i=1Bithen B

1\AB2\A:::hasB\A=S1

i=1(Bi\A), and hence B\A2`(P) soB2LA.IfA2PthenLA=`(P). Hence, ifB2`(P) thenA\B2`(P). But then this impliesLB=`(P). It follows that for allA;B2`(P), A\B2`(P).Bob HoughMath 639: Lecture 1January 24, 2017 10 / 54

Measures

Denition

Apositive measure on an algebraAis a set functionwhich satises(A)(;) = 0 for allA2AIfAi2Aare disjoint and their union is inA, then

1[ i=1A i! =1X i=1(Ai): If( ) = 1 thenis a probability measure.Bob HoughMath 639: Lecture 1January 24, 2017 11 / 54

Probability measure properties

A probability measure satises the following basic properties.(Monotonicity) IfABthen Prob(A)Prob(B).(Sub-additivity) IfAS

iAithen Prob(A)P iProb(Ai)(Continuity from below) IfA1A2:::andA=S iAithen Prob(Ai)"Prob(A)(Continuity from above) IfA1A2:::andA=T iAithen Prob(Ai)#Prob(A).Bob HoughMath 639: Lecture 1January 24, 2017 12 / 54

Atomic measures

Denition

A probability space (

;F;Prob) isnon-atomicif Prob(A)>0 implies that

there existsB2FsatisfyingBAand 0

Outer measures

Denition

Anouter measureon a measurable space (

;F) is a set function :F![0;1] satisfying (;) = 0 and(A1)(A2) for anyA1;A22FwithA1A2. (S1 n=1An)P1 n=1(An) for any countable collection of sets fAng F.Bob HoughMath 639: Lecture 1January 24, 2017 14 / 54

Outer measures

Denition

Given an outer measureon a measurable space (

;F), a setA2Fis measurable(in the sense of Caratheodory) if for each setE2F, (E) =(E\A) +(E\Ac):Bob HoughMath 639: Lecture 1January 24, 2017 15 / 54

Outer measures

Theorem

Letbe an outer measure on a measurable space(

;F). The subsetG of-measurable sets inFis a-algebra, andrestricted to this subset is a measure.See e.g. Royden pp.54{60. Bob HoughMath 639: Lecture 1January 24, 2017 16 / 54

Lebesgue measure

An outer measure on (R;2R) is given by

(A) = inf( 1X i=1b iai:A1[ i=1(ai;bi]) Lebesgue measure is obtained by restrictingto its measurable sets. The

-algebra so obtained is larger than the Borel-algebra.Bob HoughMath 639: Lecture 1January 24, 2017 17 / 54

Caratheodory's Extension Theorem

Theorem

Letbe a-nite measure on an algebraA. Thenhas a unique extension to(A).Bob HoughMath 639: Lecture 1January 24, 2017 18 / 54

Caratheodory's Extension Theorem

Proof of uniqueness.

Let1and2be two extensions ofto(A). LetA2Asatisfy

(A)<1and let

L=fB2(A) :1(A\B) =2(A\B)g:

We show thatLis a-system. SinceALandAis a-system, it then follows thatL=(A). Uniqueness then follows on taking a sequencefAngwithAn" and(An)<1.Bob HoughMath 639: Lecture 1January 24, 2017 19 / 54

Caratheodory's Extension Theorem

Proof of uniqueness.

To verify the-system property, observe

2LIfB;C2LwithCB, then

1(A\(BC)) =1(A\B)1(A\C)

=2(A\B)2(A\C) =2(A\(BC)):IfBn2LandBn"Bthen

1(A\B) = limn!11(A\Bn) = limn!12(A\Bn) =2(A\B):Bob HoughMath 639: Lecture 1January 24, 2017 20 / 54

Caratheodory's Extension Theorem

Proof of existence.

Dene set functionon(A) by

(E) = inf( 1X i=1(Ai) :E1[ i=1A i;Ai2A) Evidently(A) =(A) forA2A. Also,A2Ais measurable, since for F2(A) and >0 there existsfBig1i=1a sequence fromAsatisfyingP i(Bi)(F) +:Then (Bi) =(Bi\A) +(Bi\Ac) (F) +X i (Bi\A) +X i (Bi\Ac)(F\A) +(Fc\A): which gives the condition for measurability. Bob HoughMath 639: Lecture 1January 24, 2017 21 / 54

Caratheodory's Extension Theorem

Proof of existence.

satises the properties of an outer measure, sinceIfEFthen(E)(F)IfFS iFiis a countable union, then(F)P i(Fi). The restriction ofto its measurable sets gives the required extension of .Bob HoughMath 639: Lecture 1January 24, 2017 22 / 54

Random variables

Denition

A real valuedrandom variableon a measure space (

;F;Prob) is a functionX: !Rwhich isF-measurable, that is, for each Borel set BR, X

1(B) =f!:X(!)2Bg 2F:

Arandom vectorinRdis a measurable mapX:

!Rd.GivenA2F, the indicator function ofAis a random variable, 1

A(!) =1!2A

0!62A:Bob HoughMath 639: Lecture 1January 24, 2017 23 / 54

Random variables

Theorem

IfX1;:::;Xnare random variables andf: (Rn;BRn)!(R;B)is measurable, thenf(X1;:::;Xn)is a random variable.Theorem IfX1;X2;:::are random variables thenX1+X2+:::+Xnis a random variable, and so are inf nXn;sup nXn;limsup nXn;liminfnXn:Proof.

Exercise, or see Durrett, pp. 14{15.

Bob HoughMath 639: Lecture 1January 24, 2017 24 / 54

Distributions

Denition

Thedistributionof a random variableXonRis the probability measure on (R;B) dened by (A) = Prob(X2A):

Thedistribution functionofXis

F(x) = Prob(Xx):Bob HoughMath 639: Lecture 1January 24, 2017 25 / 54

Distributions

Theorem

Any distribution functionFhas the following properties:1Fis nondecreasing.2lim

x!1F(x) = 1;limx!1F(x) = 0:3Fis right continuous, that is,limy#xF(y) =F(x).4IfF(x) = limy"xF(y)thenF(x) = Prob(X Furthermore, any function satisfying the rst three items is the distribution function of a random variable. Bob HoughMath 639: Lecture 1January 24, 2017 26 / 54

Distributions

Proof.

All of the forward claims are straightforward.

For the reverse claim, let

= (0;1),F=Band set Prob to be Lebesgue measure. Dene

X(!) = supfy:F(y)< !g:

Then f!:X(!)xg=f!:!F(x)g;

which follows by the right-continuity ofF. Hence Prob(Xx) =F(x).Bob HoughMath 639: Lecture 1January 24, 2017 27 / 54

Distributions

Denition

IfXandYinduce the same distributionon (R;B), we sayXandYare equal in distribution. We writeX=dY.Denition When the distribution functionF(x) = Prob(Xx) has the form

F(x) =Z

x 1 f(y)dy we say thatXhasdensity functionf.Bob HoughMath 639: Lecture 1January 24, 2017 28 / 54

Example distributions

Uniform distribution on (0,1). Densityf(x) = 1 forx2(0;1) and 0 otherwise.Exponential distribution with rate. Densityf(x) =exforx>0,

0 otherwise.Standard normal distribution. Densityf(x) =exp

x22 p2:Bob HoughMath 639: Lecture 1January 24, 2017 29 / 54

Example distributions

Uniform distribution on the Cantor set. Dene distribution functionF byF(x) = 0 forx0,F(x) = 1 forx1,F(x) =12 forx2[13 ;23

F(x) =14

forx2[19 ;29 ],F(x) =34 forx2[79 ;89 ],....Point mass at 0. The distribution function hasF(x) = 0 forx<0, F(x) = 1 forx0.Lognormal distribution. LetXbe a standard Gaussian variable. exp(X) is lognormal.Chi-square distribution. LetXbe a standard Gaussian variable.X2 has a chi-squared distribution. Bob HoughMath 639: Lecture 1January 24, 2017 30 / 54 Example distributions onZBernoulli distribution, parameterp. Prob(X= 1) =p, Prob(X= 0) = 1p.Poisson distribution, parameter.Xis supported onZand Prob(X=k) =ekk!.Geometric distribution, success probabilityp2(0;1).Xis supported onZand Prob(X=k) =p(1p)k1, fork= 1;2;:::.Bob HoughMath 639: Lecture 1January 24, 2017 31 / 54

Integration

The Lebesgue integral against a-nite measure is dened as usual for1Simple functions

2Bounded functions

3Nonnegative functions

4General functions

Bob HoughMath 639: Lecture 1January 24, 2017 32 / 54

Integral inequalities

Theorem (Jensen's inequality)

Letbe convex onR. Ifis a probability measure, andfand(f)arequotesdbs_dbs19.pdfusesText_25
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