Lecture1:Ideas from measure theorySTAT205Lecturer:Jim Pitman Scribe:Sivakumar Rathinam<rsiva@>1.1Probability spacesThis lecture introduces some ideas from measure theory which are the foundation of the modern theory of probability.The notion of a probability space is defined,and Dynkin’s form of the monotone class theorem is presented.Definition1.1LetΩbe a set of pointsω.In probability theory,Ωrepresents all possible outcomes of an experiment or observation.Example1.2Tossing a coin has a set of outcomesΩ={Head,T ail}.Example1.3Position of a body in a3-D Euclidean space belongs to the setΩ=R3.A subset ofΩis 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 define classes of subsets of the spaceΩ.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 definitions would be in this regard.Once we define the classes of sets we are interested in,one can assign a probability measure to each of these sets.Definition1.4A class F of subsets of a spaceΩis called afield if it containsΩitself and is closed under complements andfinite unions.That is1.Ω∈F2.A∈F implies A c∈F3.A,B∈F implies A∪B∈F1-1Note 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 definition can be replaced with A,B∈F implies A∩B∈F.Definition1.5A class F of subsets ofΩis aσ-field if it is afield and if it is closed under the formation of countable unions.That is,1.F is afield.2.A1,A2,...∈F implies A1∪A2∪....∈F.Afield is closed underfinite set theoretic operations whereas aσ-field is closed under countable set theoretic 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 define a class denoted byσ(A).This class is called theσ-field generated by A.It is defined as the intersection of all theσ-fields containing A(or the smallestσ-field containing A).Now,we give the definition of the probability measure.Definition1.6A set function1P on aσ-field F is a probability measure if it satisfies the following conditions:1.0≤P(A)≤1for A∈F.2.P(∅)=0,P(Ω)=1.3.If A i∈F is a countable union of sets,then i A i∈F.If F is aσ-field,then the triple(Ω,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 afield 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,thenP ∞ k=1A k ≤∞ k=1P(A k).(1.1)Example1.8If A is the class of subintervals ofΩ=(0,1),then the sigmafield 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 defined as(Ω,F,P),whereΩ=(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<b.Now we will present a key result that would help us to extend the results we have on afield A to theσ-field generated by A.This is stated as the Identitification Lemma for Probabilities:Lemma1.9(Identification 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.Definition1.10A collection of subsets D of setΩis called aλ-system if1.Ω∈D2.If A∈B and B∈D,A⊂B⇒B−A∈D3.If A n∈D and A n↑A⇒A∈DTheorem1.11(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 identification lemma on probabilities.Proof:(of Lemma1.9)Consider D={A∈σ(A):P(A)=Q(A)}.Let’s check thatD is aλ-system.1.Ω∈D because P(Ω)=Q(Ω)=1.2.Let A∈D,B∈D.Then A⊂B implies B−A∈D.This is becauseP(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.。