Lebesgue Integration - Proof Techniques

Proof Techniques

To illustrate some of the proof techniques used in Lebesgue integration theory, we sketch a proof of the above mentioned Lebesgue monotone convergence theorem. Let {f_k}_{k ∈ N} be a non-decreasing sequence of non-negative measurable functions and put

By the monotonicity property of the integral, it is immediate that:

and the limit on the right exists, since the sequence is monotonic. We now prove the inequality in the other direction. It follows from the definition of integral that there is a non-decreasing sequence (g_n) of non-negative simple functions such that g_n ≤ f  and

Therefore, it suffices to prove that for each n ∈ N,

We will show that if g is a simple function and

almost everywhere, then

By breaking up the function g into its constant value parts, this reduces to the case in which g is the indicator function of a set. The result we have to prove is then

Suppose A is a measurable set and {f_k}_{k ∈ N} is a nondecreasing sequence of non-negative measurable functions on E such that

for almost all x ∈ A. Then

To prove this result, fix ε > 0 and define the sequence of measurable sets

By monotonicity of the integral, it follows that for any k ∈ N,

Because almost every x will be in B_k for large enough k, we have

up to a set of measure 0. Thus by countable additivity of μ, and since B_k increases with k,

As this is true for any positive ε the result follows.