Haven’t posted in two months! Time for some more measure theory, I suppose. Let’s start with Fatou’s lemma, which is a very useful tool:

Theorem 1Let be a measure space and a sequence of nonnegative measurable functions. Then

*Proof:* Since , this inequality is preserved by the integral and hence right-hand side is a monotone increasing sequence of real numbers, which has a limit in . For any , we have for , and hence

and by monotone convergence,

Hence

Note that this inequality requires no assumptions on (unlike monotone convergence). An immediate application is the so-called Dominated Convergence Theorem, but first, a digression. Given a measure space , we define

From the definition of it is clear that if and only if the integrals of , , , and are finite. For we define

i.e. if and only if

Now, we would like to define a norm on by e.g.

The problem is that this is only a pseudonorm, as , but also for a.e. To remedy this, we consider the equivalence relation on where

Then we define

to be the set of equivalence classes of under . That is, we consider functions that agree except on a set of measure zero (which does not change the integral) to be equivalent. Then defines a norm on , and a metric. We will discuss spaces in more detail later. Back to the Dominated Convergence Theorem:

Theorem 2Let be a sequence of measurable functions with pointwise and suppose there exists a function such that for all . Then and

*Proof:* Since for all , letting we have , so

and hence . Now by Fatou’s lemma,

and

Since we conclude that

and hence by another application of Fatou’s lemma,