Sheng Li (gmachine1729) wrote,
Sheng Li
gmachine1729

Monotone convergence theorem, Fatou’s lemma, and dominated convergence theorem

Originally published at 狗和留美者不得入内. You can comment here or there.

In undergrad, I learned these three major theorems for Lebesgue integration but never understood them. I was never able to actually re-prove them independently. However, I succeeded now with all three after reading the gist of them a week or two ago. I don’t them I will ever forget them (by which I mean I will always be able to reconstruct them), though that is of course much easier said than done. I will write them up now to confirm.

Proposition 1 Assuming countable additivity of [公式] , [公式] Proof: We leave this to the reader.

Proposition 2 For a real measurable function [公式] and any real number [公式] , the following are equivalent.

  1. [公式] is measurable.
  2. [公式] is measurable.
  3. [公式] is measurable.
  4. [公式] is measurable.

Proof: Left to the reader.

Proposition 3 For any Lebesgue measurable non-negative function [公式] ,[公式] is countably additive on Lebesgue measurable sets.

Proof: We leave this to the reader.

Theorem 1 (Monotone convergence theorem) Let [公式] be a sequence of measurable functions such that [公式] for all [公式] and [公式] pointwise on set [公式] . Then, [公式] is measurable, with

[公式]

Proof: The [公式] inequality is obvious from monotonicity. The hard part is developing a technique to prove the other way round. For that, the heuristic is to take a monotonic sequence converging for the right hand side and associate it directly with a sequence converging to the right hand side that is bounded above by with it at every index. For the right hand side, the natural choice is a sequence of simple functions [公式] such that

[公式]

which exists by definition of Lebesgue integral. For any fixed [公式] , we can let

[公式] The definition of simple function along with Proposition 2 tells us that [公式] is measurable for all [公式] . [公式] since by pointwise convergence, for every [公式] , there exists an [公式] such that [公式]. By Proposition 3, we have for all [公式]

[公式] Taking the limit of this gives us

[公式]

Since this holds for all [公式] , it must hold for [公式] as well, which completes our proof. [公式]

Lemma 1 (Fatou’s lemma) For a sequence of measurable non-negative functions [公式] ,

[公式] Proof: By definition of [公式] , [公式] for all [公式] , from which follows

[公式]

[公式] is the pointwise limit of a monotonically non-decreasing sequence. Thus, by the monotone convergence theorem along with taking the limit of [公式] ,

[公式]

This completes our proof. [公式]

Theorem 2 (Dominated converged theorem) Suppose sequence of measurable functions [公式] pointwise and there exists measurable [公式] such that [公式] for all [公式] . Then [公式] is measurable and

[公式]

Proof: By the monotone convergence theorem and Fatou’s lemma, we have

[公式]

Seeing that proving

[公式]

suffices via the squeeze theorem, we also notice that

[公式] which gives

[公式]

Multiplying this by [公式] yields the lower bound of [公式] , which completes our proof. We note that in this proof we have in some sense implicitly assumed decomposition of [公式] into [公式] and [公式]. [公式]

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