Chapter 19 Abstract Measures And Measurable Functions

The Lebesgue integral is much better than the Rieman integral. This has been known for over 100 years. It
is much easier to generalize to many dimensions and it is much easier to use in applications. That is why
I am going to use it rather than struggle with an inferior integral. It is also this integral which is most
important in probability. However, this integral is more abstract. This chapter will develop the abstract
machinery necessary for this integral.

The next definition describes what is meant by a σ algebra. This is the fundamental object 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 2^{Ω} but I won’t refer to it this
way.

Definition 19.0.1ℱ ⊆P

(Ω )

, the set of all subsets of Ω, is called a σ algebra if it contains ∅,Ω,and is closed with respect to countable unions and complements. That is, if

{A }
n

_{n=1}^{∞}is countableand each A_{n}∈ ℱ, then ∪_{n=1}^{∞}A_{m}∈ ℱ also and if A ∈ ℱ, then Ω ∖ A ∈ ℱ. It is clear that anyintersection of σ algebras is a σ algebra. If K⊆P

(Ω)

, σ

(K)

is the smallest σ algebra which containsK.

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} = ∪_{k}F_{k}^{C}∈ℱ and so

( )C ( )
∩kFk = (∩kFk)C = ∪kFkC C ∈ ℱ.

Example 19.0.2You could consider ℕ and for your σ algebra, you could have P

(ℕ)

. This satisfiesall the necessary requirements. Note that in fact, P

(S)

works for any S. However, useful examplesare not typically the set of all subsets.