Monotone Convergence

Let be a non-negative Measurable Function.
Let be a sequence of such functions.
Suppose that .
Then also the Integrals .

Proof

Need to check all cases but simple enough for each

Case 1

and

Case 2

Simple Function,

Case 3

Simple Function, Simple Function

Case 4

Simple Function, Measurable Function

Case 5

Measurable Function, Measurable Function, both

Proposition

Let be a sequence of nonnegative Measurable Functions.
Then: