Measure Theory/Properties of Nonnegative Integrals

Properties of Nonnegative Integrals
Throughout this lesson you may assume that


 * $$E\in\mathcal M$$
 * $$f,g:E\to\Bbb R^{\ge 0}$$ are measurable, nonnegative functions
 * $$c,d\in\Bbb R$$

Recall the definition of a nonnegative integral,


 * $$\int_E f = \sup\int_E h$$ where the supremum is taken over all functions $$h:E\to \Bbb R$$ which are bounded and $$\lambda(\{x\in E: h(x)\ne 0\})<\infty$$.

Several of the basic properties of nonnegative integrals are familiar from the lesson on bounded integrals. Consistency, linearity, order-preserving.

Notice that it makes no sense to talk about theorems which have to do with absolute values, such as the triangle inequality, since the function and therefore the integral are nonnegative, and hence equal to their own absolute value.

We then prove what might debatably be the most essential theorem in all of measure theory, the monotone convergence theorem. It will be helpful to first prove Fatou's lemma.

It will turn out that in later studies, Fatou's lemma takes on a life of its own, so to speak.

Consistency
We yet again have conflicting definitions of the integral. This time, if $$\lambda(E)<\infty$$ and $$f:E\to\Bbb R$$ is bounded and measurable and nonnegative, then $$\int_E f$$ could mean either the bounded integral or the nonnegative integral. In this subsection, we will understand that $$\int_E f$$ refers to the nonnegative integral, and therefore we use the notation $$(B)\int_E f$$ to refer to the bounded integral.

Exercise 1. Bounded-Nonnegative Consistency
Prove that $$\int_Ef = (B)\int_E f$$ for any function f as described above. The style of proof should be very similar to that used for Prove Simple-Bounded Consistency.

Exercise 1. Prove Linearity and Order Preserving
Prove linearity and order preserving. The proof should be very similar in style to the one given for bounded integrals.

Fatou's Lemma
Let $$\langle f_n\rangle$$ be a sequence of nonnegative measurable functions and $$f_n\overset{p.w.}\to f$$ on a set E, then Fatou's lemma states


 * $$\int_E f\le\underline{\lim}\int_E f_n$$

It suffices to prove that $$\underline \lim \int_E f_n$$ is an upper bound on the set of all $$\int_E h$$ where $$h\le f$$ is bounded, and is nonzero only a finite-measure set.

For any such h, we will construct a new sequence of functions which runs somewhat "in parallel" with the sequence $$f_n$$.


 * $$h_n=\min\{h,f_n\}$$

so that therefore $$h_n\le f_n$$ and therefore $$\int_E h_n\le \int_E f_n$$ and therefore $$\underline\lim\int_E h_n\le\underline\lim\int_E f_n$$.

Exercise 2. Why Liminf?
In the last step above, we took the limit inferior of both sides of an inequality. Why did we not simply take the limit?

Hint: It has to do with the fact that the sequence of infima is a necessarily increasing sequence, for any "source" sequence. It also has to do with the fact that each term in this particular sequence is necessarily nonnegative.

Exercise 3. Prove Fatou's
Prove Fatou's lemma by the following steps.

1. Prove that for any fixed $$x\in E$$ the limit $$\lim_{n\to\infty}h_n(x)=h(x)$$.

2. Apply the Bounded Convergence Theorem.

3. Use the fact that when a limit exists, it must equal its limit inferior.

Monotone Convergence Theorem
The monotone convergence theorem has the same set of assumptions as Fatou's lemma, but then further imposes the assumption that the sequence is monotonically increasing:


 * $$f_n\le f_{n+1}$$ for each $$1\le n$$

It then states that therefore the swaparoo follows:


 * $$\lim_{n\to\infty}\int_Ef_n = \int_E f$$

Exercise 4. Prove the MCT
Prove the MCT. An immediate and instinctive application of Fatou's lemma is a great first step.

Now use the fact that the sequence is increasing, to infer that


 * $$\overline\lim\int_Ef_n\le\int_E f$$

and then conclude the theorem.