Another Proof of the Levi's Theorem

The Levi’s Theorem, also known as Levi’s monotone convergence theorem, shows that for a non-negative measurable function, the order of the limit operator and integral operator can be rearranged. It is stated as follows:

Suppose and are non-negative measurable functions on measurable set D. And for almost all , is a monotone increasing sequence that converges to , then

In Real Variable Funtions, a book written by Xingwei, Zhou, the proof of the theorem is beautiful. Let me show you how it is done. Since the Lebesgue integral of a non-negative measurable function is defined by the limit of the integral of a monotone increasing sequence of non-negative simple functions. Every can be approximated by a monotone increasing sequence of non-negative simple functions, which are denoted as . The next step is to construct a new function series like this:

To write it down,

Therefore, the sequence have the following properties:
(1) is monotone increasing
(2)

In the second property, we first let , then let , so the limit of becomes . Besides, according to the first property, the integral of can be defined by the limit of the integrals of .

The idea behind the proof, whose key point lies in the construction of , is inspiring. In fact, ways to construct is not unique. It works as long as the sequence you constructed satisfies the two properties. I found another way to build the sequence: