Contents I Topology , Continuity , And Algebra 9

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 but I won’t refer to it this way. Definition 6.0.1 F ⊆ 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 {An}n=1 is countable and each An ∈ F , then ∪n=1Am ∈ 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. 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 = ∪kF k ∈ F and so ∩kFk = ( (∩kFk) )C = ( ∪kF k )C ∈ F . Example 6.0.2 You could consider N and for your σ algebra, you could have P (N). This satisfies 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. A useful idea is the idea of distance from a point to a set. Definition 6.0.3 Let (X, d) be a metric space and let S be a nonempty set in X. Then dist (x, S) ≡ inf {d (x, y) : y ∈ S} . The following lemma is the fundamental result. Lemma 6.0.4 The function, x → dist (x, S) is continuous and in fact satisfies |dist (x, S)− dist (y, S)| ≤ d (x, y) .