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
Definition 7.0.1 ℱ ⊆P
If ℱ is a σ algebra, then it is also closed with respect to countable intersections. Here is why. Let
Example 7.0.2 You could consider ℕ and for your σ algebra, you could have P
A useful idea is the idea of distance from a point to a set.
Definition 7.0.3 Let
The following lemma is the fundamental result.
Proof: Suppose dist