Chapter 9

Measures and Measurable FunctionsThe Lebesgue integral is much better than the Rieman integral. This has been known forover 100 years. It is much easier to generalize to many dimensions and it is much easier touse in applications. It is also this integral which is most important in probability. However,this integral is more abstract. This chapter will develop the abstract machinery for thisintegral.

The next definition describes what is meant by a σ algebra. This is the fundamentalobject which is studied in probability theory. The events come from a σ algebra of sets.Recall that P (Ω) is the set of all subsets of the given set Ω. It may also be denoted by 2Ω

but I won’t refer to it this way.

Definition 9.0.1 F ⊆P (Ω) , the set of all subsets of Ω, is called a σ algebra ifit contains /0,Ω, and is closed with respect to countable unions and complements. Thatis, if {An}∞

n=1 is countable and each An ∈F , then ∪∞n=1An ∈F also and if A ∈F , then

Ω\A ∈F . It is clear that any intersection of σ algebras is a σ algebra. If K ⊆P (Ω) ,σ (K ) is the smallest σ algebra which contains K . In fact, the intersection of all σ

algebras containing K is obviously a σ algebra so this intersection is σ (K ).

If F is a σ algebra, then it is also closed with respect to countable intersections. Here

is why. Let {Fk}∞

k=1 ⊆ F . Then (∩kFk)C = ∪kFC

k ∈ F and so ∩kFk =((∩kFk)

C)C

=(∪kFC

k

)C ∈F .

Example 9.0.2 You could consider N and for your σ algebra, you could have P (N). Thissatisfies all the necessary requirements. Note that in fact, P (S) works for any S. However,useful examples are not typically the set of all subsets.

9.1 Simple Functions and Measurable FunctionsA σ algebra is a collection of subsets of a set Ω which includes /0,Ω, and is closed withrespect to countable unions and complements.

Definition 9.1.1 A measurable space, denoted as (Ω,F ) , is one for which F is aσ algebra contained in P (Ω). Let f : Ω→ X where X is a metric space. Then f is said tobe measurable means f−1 (U) ∈F whenever U is open.

It is important to have a theorem about pointwise limits of measurable functions. Thefollowing is a fairly general such theorem which holds in the situations to be consideredin this book. First recall dist(x,S) in Lemma 3.12.1 which impliles that x→ dist(x,S) iscontinuous.

Theorem 9.1.2 Let { fn} be a sequence of measurable functions mapping Ω to themetric space (X ,d) where (Ω,F ) is a measureable space. Suppose the pointwise limitf (ω) = limn→∞ fn (ω) for all ω. Then f is also a measurable function.

Proof: It is required to show f−1 (U) is measurable for all U open. Let

Vm ≡{

x ∈U : dist(x,UC)> 1

m

}.

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