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lecture1

Lecture1:Ideas from measure theory

STAT205Lecturer:Jim Pitman Scribe:Sivakumar Rathinam

1.1Probability spaces

This lecture introduces some ideas from measure theory which are the foundation of the modern theory of probability.The notion of a probability space is de?ned,and Dynkin’s form of the monotone class theorem is presented.

De?nition1.1Let?be a set of pointsω.In probability theory,?represents all possible outcomes of an experiment or observation.

Example1.2Tossing a coin has a set of outcomes?={Head,T ail}.

Example1.3Position of a body in a3-D Euclidean space belongs to the set?=R3.

A subset of?is called an event.It is natural to ask questions like whether an outcome of a random experiment belongs to to a event or not.To do this,we need to de?ne classes of subsets of the space?.Also,since we would be talking about any combination of events,a systematic treatment would require the class of sets to have the necessary set theoretic operations:namely the sets being closed under countable unions and intersections.The next few de?nitions would be in this regard.Once we de?ne the classes of sets we are interested in,one can assign a probability measure to each of these sets.

De?nition1.4A class F of subsets of a space?is called a?eld if it contains?itself and is closed under complements and?nite unions.That is

1.?∈F

2.A∈F implies A c∈F

3.A,B∈F implies A∪B∈F

1-1

Note that by DeMorgan’s law,given that F is closed under complement,F is closed under unions if and only if F is closed under intersections.Therefore,A,B∈F implies A∪B∈F in the above de?nition can be replaced with A,B∈F implies A∩B∈F.

De?nition1.5A class F of subsets of?is aσ-?eld if it is a?eld and if it is closed under the formation of countable unions.That is,

1.F is a?eld.

2.A1,A2,...∈F implies A1∪A2∪....∈F.

A?eld is closed under?nite set theoretic operations whereas aσ-?eld is closed under countable set theoretic https://www.doczj.com/doc/1e7527905.html,ually in a problem dealing with probabilities, one is dealing with a small class of subsets A,for example the class of subintervals of(0,1].It is possible that when we perform countable operations on such a class A of sets,we might end up operating on sets outside the class A.Hence,we would like to de?ne a class denoted byσ(A).This class is called theσ-?eld generated by A.It is de?ned as the intersection of all theσ-?elds containing A(or the smallestσ-?eld containing A).Now,we give the de?nition of the probability measure.

De?nition1.6A set function1P on aσ-?eld F is a probability measure if it satis?es the following conditions:

1.0≤P(A)≤1for A∈F.

2.P(?)=0,P(?)=1.

3.If A i∈F is a countable union of sets,then i A i∈F.

If F is aσ-?eld,then the triple(?,F,P)is called a probability measure space or simply a probability space.The countable additivity of the probability measure gives rise to the following properties that are stated in a theorem.

Theorem1.7Let P be a probability measure on a?eld F.

1.Continuity from below:If A n and A lie in F and A n↑A,then P(A n)↑P(A).

2.Continuity from above:If A n and A lie in F and A n↓A,then P(A n)↓P(A).

3.Countable subadditivity:If A1,A2...and ∞k=1A k lie in F,then

P ∞ k=1A k ≤∞ k=1P(A k).(1.1)

Example1.8If A is the class of subintervals of?=(0,1),then the sigma?eld generated by A,denoted by B,is called the collection of Borel sets of the unit interval. The probability space on a unit interval is then de?ned as(?,F,P),where?=(0,1), F={A∩(0,1):A∈B}and P(B)=λ(B)for B∈F.Hereλis the Lebesgue measureλ((a,b])=b?a for a

Now we will present a key result that would help us to extend the results we have on a?eld A to theσ-?eld generated by A.This is stated as the Identiti?cation Lemma for Probabilities:

Lemma1.9(Identi?cation Lemma for Probabilities)Let P and Q be two prob-ability measures onσ(A)where A is closed under intersections.If P(A)=Q(A)for A∈A,then P(A)=Q(A)for all A∈σ(A).

Before we prove this lemma,let’s state some basic tools from measure theory that are very useful.

De?nition1.10A collection of subsets D of set?is called aλ-system if

1.?∈D

2.If A∈B and B∈D,A?B?B?A∈D

3.If A n∈D and A n↑A?A∈D

Theorem1.11(Dynkin’sπ-λTheorem)Suppose A is a collection of sets closed under∩(aπ-system).If D is aλ-system with A?D,thenσ(A)?D.

With this theorem let’s try to prove the identi?cation lemma on probabilities.

Proof:(of Lemma1.9)Consider D={A∈σ(A):P(A)=Q(A)}.Let’s check that

D is aλ-system.

1.?∈D because P(?)=Q(?)=1.

2.Let A∈D,B∈D.Then A?B implies B?A∈D.This is because

P(B)=Q(B)?P(B?A)+P(A)=Q(B?A)+Q(A)?P(B?A)=Q(B?A).

3.A n∈D and A n↑A implies A∈D.This is because P(A n)=Q(A n) P(A)=

Q(A)using theorem1.7.

Now directly applying Dynkinsπ?λTheorem,we get A?D impliesσ(A)?D.

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