Lecture 6. The Dynkin ? ? ? Theorem. It is often the case that two
We shall do the ? ? ? theorem and use it in the case of Lebesgue Proof. Let µ be a translation-invariant Borel measure on R which assigns finite.
Dynkins ?-? Theorem
15 nov. 2016 Exercise 1. Prove each of the following statements. I. Every ?-algebra is both a ?-system and ?-system. II.
Lecture 8 ?-systems ?-systems
https://mpaldridge.github.io/teaching/ma40042-notes-08.pdf
Math 639: Lecture 1 - Measure theory background
24 janv. 2017 Dynkin's ? ? ? Theorem. Lemma. Let L be a ?-system which is closed under intersection. Then L is a ?-algebra. Proof.
Dynkins ?-? Theorem: If ?? ?for a ?-system ?and a ?-system ?
Proof: Define ?0 to be the smallest ?-system containing ?. Then by definition. ?? ?0 ? ?. If we can show that ?0 is a ?-field
Dynkin (?-) and ?-systems; monotone classes of sets and of functions
7 févr. 2018 Theorem 6. Let L be a ?-sytem on ?. Then ??(L) = ??(L). Proof. ... Corollary 7 (?-? theorem/Dynkin's lemma/Sierpinski class theorem).
- Theorem
? -system. Proof of the claim 1: For each A (in L or in ) let. 2X.
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Real Analysis Oral Exam study notes
1.3. PI-LAMBDA THEOREM. 6. Proof. For any arbitarty sequence An ? C we can create Bn ? C which are disjoint with ?n k=1Bk = ?n.
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Lecture 6 The Dynkin ? ? Theorem - LSU
6 1 Dynkin’s ? ?? Theorem Let P be a ?-system of subsets of Xand L a ?-system of subsets of X Suppose also that P ? L Then : ?(P) ? L i e L contains the ?-algebra ?(P) generated by P We will do the proof later but let us apply it to prove the uniqueness of Lebesgue measure 49
problem solving - Why use dimensionless variables - Mathematics Stack
TheCarathéodory extension theorem allows us to de?ne a measure explicitly for only a small collection of simplesets which may or may not form a?-algebra and automatically extend the measure to a proper measurablespace The uniqueness claim in the extension theorem makes use ofDynkin’s?-?theorem
Front Page Statistical Science
every ?-systems which contains (same proof as the construction of generated ?-filed which based mainly on the facts that an element is in the intersection iff it is in every sets) In particular because L is one of the set in the intersection (So we have ) P LL0 ? PLL??0 Claim 1: L0 is a ?-system Proof of the claim 1:
Real Analysis Oral Exam study notes Notes transcribed by
1 3 PI-LAMBDA THEOREM 6 Proof orF any arbitarty sequence A n2C we can create B n2Cwhich are disjoint with [n k=1 B k= [n k=1 A k by doing intersections (ok since Cis a ? system) and complements (ok since Cis a -system) Then since Cis a system we have that [1 k=1 B k2Cand so [1 k=1 A k= [1 k=1 B k2Ctoo! Theorem
Claim 1 The ?-system completion of a ? ? ? ? ? ? AB
The ?-system completion of a ?-system is itself a ?-system Combined with the comparatively trivial fact that a ?-system that is also a ?-system is a ?-algebra this concise statement is actually enough to prove the usual ?-? theorem Let ? be a ?-system and ? be the generated ?-system
Searches related to pi lambda theorem proof filetype:pdf
Created Date: 2/17/2008 12:29:26 PM
What is the Pi theorem?
- The Pi theorem gives you a procedure to determine the dimensionless groups. So much work has come out of using non dimensional analysis. If you want refer to G.I.Barenblatt (Scaling, Self-similarity, and Intermediate Asymptotics) book or another classic from an application point of view is Sedov (Similarity and Dimensional methods in mechanics).
What is Lami's theorem?
- Lami's Theorem is applied in a static analysis of structural and mechanical systems. Lami's Theorem is named after Bernard Lamy. Lami 's Theorem states, "When 3 forces related to the vector magnitude acting at the point of equilibrium, each force of the system is always proportional to the sine of the angle that lies between the other two forces."
How can I prove that I have connectivity from the lambda function?
- I can prove that I have connectivity from the lambda function to the internet by setting up a Socket connection e.g. Socket s = new Socket(InetAddress.getByName("bbc.co.uk"), 80);and I can retrieve data this way.
What is Pi Lambda Phi?
- The Pi chapter of the national fraternity Pi Lambda Phi was established at Dartmouth College in 1924. The membership of the Dartmouth chapter was predominantly Jewish.
Math 639: Lecture 1
Measure theory background
Bob Hough
January 24, 2017
Bob HoughMath 639: Lecture 1January 24, 2017 1 / 54Probability 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 / 54Algebras 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 aiDenition
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.DenitionA-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.DenitionIn the case that
has a topologyTof open sets, theBorel-algebrais (T).Bob HoughMath 639: Lecture 1January 24, 2017 5 / 54Borel-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 / 54Dynkin'sTheoremDenition
A-systemis a collectionPof sets closed under nite intersections. A -systemis a collectionLof sets satisfying the following2LFor anyA;B2LsatisfyingAB,BnA2LIfA1A2:::is a sequence fromLandA=S1
i=1AithenA2L.Bob HoughMath 639: Lecture 1January 24, 2017 7 / 54Dynkin'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 / 54Dynkin'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. I2LAsinceA2`(P)
IIfB;C2LAandBC, then
A\(BC) = (A\B)(A\C)2`(P).
IIfB1B2:::is a sequence fromLAwithB=S1
i=1Bithen B1\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 / 54Measures
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 / 54Probability 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 / 54Atomic measures
Denition
A probability space (
;F;Prob) isnon-atomicif Prob(A)>0 implies thatthere existsB2FsatisfyingBAand 0 -algebra so obtained is larger than the Borel-algebra.Bob HoughMath 639: Lecture 1January 24, 2017 17 / 54 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 which follows by the right-continuity ofF. Hence Prob(Xx) =F(x).Bob HoughMath 639: Lecture 1January 24, 2017 27 / 54Outer 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 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 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; 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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