Convergence Theorems

For all statements on this page, assume that is a measure space.

Monotone Convergence Theorem

For each let be a measurable function. If almost everywhere, then

Note that the pointwise limit always exists and is measurable by this proposition.

Fatou’s Lemma

For each let be a nonnegative measurable function. Then

In the following proof we omit and for visual clarity.

Proof   By definition, we have , where . Now is a monotonic sequence of nonnegative measurable functions. By the Monotone Convergence Theorem

For all one has , hence by the monotonicity of the integral. This implies

for all . In the limit we obtain

thereby completing the proof.

Dominated Convergence Theorem

Let be a measure space. For each let (or ) be a measurable function. Suppose that the pointwise limit exists almost everywhere. Suppose further that there exists an integrable function such that almost everywhere for all . Then the functions and are all integrable, and

Proof   TODO