monotone convergence theorem (integral version)

This is a way to interchange the limit of an integral with the integral of the limit (under certain conditions, since that doesn’t always work).

Theorem

For a measure space and is an increasing sequence of measurable functions, let such that

Then

Proof

(Before anything else, we know that the function is measurable)

First, prove . Each thus since Lebesgue integral preserves order .

Next, prove .

We need a little space, so first let be a simple measurable function.

Choose and consider the set

Since then so it’s an increasing sequence of sets and since we “shrank” the function so .

So we use these constructed sets along with the sets from .

This lets us use these in

We need these which is an increasing sequence of sets to move the limit inside the measure. Integrating the above gives

By the continuity of measure, along with the sets

So the limit plays nice in equation (1), so we take the limit,

Now take the supremum over all the partitions (aka let the simple function become a better and better approximation)

Then just let get arbitrarily close to 1. And with that we have both sides so