Measure Theory/Properties of General Integrals

Properties and Convergence of General Integrals
Recall the definition of the length-measure integral of a measurable function $$f:E\to\Bbb R$$,


 * $$ \int_E f = \int_Ef^+-\int_Ef^-$$

and recall the definitions


 * $$f^+=\max\{f,0\},\quad f^-=\max\{-f,0\}$$

In this lesson, assume


 * $$ f,g,f_n:E\to\Bbb R$$ are measurable functions
 * $$ f_n\to f $$
 * $$ c,d\in\Bbb R$$
 * $$ |f_n|\le g $$ for all $$n\in\Bbb Z^+$$

Exercise 1. Consistency
Explain why the check for consistency, which we have done for previous generalizations of integral definitions, is trivial in this case.

Exercise 2. Basic Properties of General Integrals
Prove the basic properties of general integrals: Linearity, order-preserving, triangle inequality, the ML bound, and finite additivity.

Note: What is the positive part of $$f+g$$ in terms of $$f^+,f^-,g^+,g^-$$?

Lebesgue's Dominated Convergence Theorem
Here resides the last (or, depending on how you count, the penultimate) of the great and famous convergence theorems of measure theory.

Besides the assumptions at the top of this page, further assume that g is integrable.

Lebesgue's Dominated Convergence Theorem then states that the swaparoo follows.


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

Exercise 3. Prove the LDCT
 Part A. With the observation that $$g-f_n$$ is a nonnegative function, use Fatou's lemma.

 Part B. Use an earlier result to establish that f is integrable. Then infer from Part A. that $$\int_Eg-\int_E f\le\int_Eg - \overline\lim\int_E f_n$$.

Part C. Now apply reasoning similar to that in parts A. and B. above to $$g+f_n$$ to infer $$\int_E f\le\underline\lim\int_Ef_n$$ and conclude the proof.

Part D. Prove the following generalization of the LDCT. Let $$\langle g_n\rangle$$ be a sequence of integrable functions. Assume further that * $$\langle g_n\rangle$$ so-to-speak "pairwise dominates" the sequence $$\langle f_n\rangle$$. Formally this means $$|f_n|\le g_n$$ for each $$n\in\Bbb Z^+$$. * $$ g_n\to g$$ on E. * $$ \lim_{n\to\infty}\int_Eg_n = \int_E\lim_{n\to\infty}g_n = \int_E g$$ Now prove that $$\lim_{n\to\infty}\int_Ef_n=\int_E\lim_{n\to\infty}f_n=\int_Ef$$. The proof should merely reiterate all of the proof of the LDCT, but replacing g by $$g_n$$ where appropriate.