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. Complex analysis does not really require the use of this superior integral however, but the approach we take here will involve integration of a function of more than one variable and when you do this, the Riemann integral becomes totally insufferable, forcing one to consider things like the Jordan content of the boundary and so forth. It is much better to go ahead and deal with the better integral even though it is more abstract.
Definition 5.0.1 Let Ω be a nonempty set. ℱ ⊆P
Observation 5.0.2 If you have sets Ai ∈ℱ a σ algebra, then
Thus countable unions, intersections and complements of sets of ℱ stay in ℱ.