A proof of Monotone Convergence Theorem:
Suppose an increasing sequence a.e., then it is immediate that . The theorem claims the reverse inequality holds as well. To show this, it's sufficient to show that
for a simple function with (since, by definition, .)
To this end, fix , we consider the sets
Since is increasing, . Also, for all n
Now we notice that is a positive measure on X. Thus, by the so-called "continuity from below" property of measures,
Together with the previous estimate, this implies
By virtue of being simple, taking the supremum of gives
Taking the supremum over proves the claim.
Notice the argument hinges on how one partitions the range (as oppose to the domain in the Riemann case) of the functions into suitable parts, passes back to the preimages, and then applies the machinery of measure theory.
The assumption that the sequence be increasing is needed, for the areas of might "escape to infinity" and equality may fail. In the absence of this assumption, taking the infimum and applying the monotone convergence theorem yields the Fatou's lemma:
We note that, assuming Fatou's lemma, the monotone convergence theorem follows trivially: