Measure Theory/Properties of General Integrals

Properties and Convergence of General Integrals

Recall the definition of the length-measure integral of a measurable function $f:E\to {\mathbb {R}}$,

$\int _{E}f=\int _{E}f^{+}-\int _{E}f^{-}$

and recall the definitions

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

In this lesson, assume

• $f,g,f_{n}:E\to {\mathbb {R}}$ are measurable functions
• $f_{n}\to f$
• $c,d\in {\mathbb {R}}$
• $|f_{n}|\leq g$ for all $n\in {\mathbb {Z}}^{+}$

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.

Definition: dominated sequence

Let $\langle f_{n}\rangle$ be a sequence of functions $f_{n}:A\to {\mathbb {R}}$ and $g:A\to {\mathbb {R}}$ another function. If for all $n\in {\mathbb {Z}}^{+}$ we have $|f_{n}|\leq g$ then we say that g dominates the sequence $\langle f_{n}\rangle$.

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 $\int _{E}g-\int _{E}f\leq \int _{E}g-{\overline {\lim }}\int _{E}f_{n}$.

Part C.
Now apply reasoning similar to that in parts A. and B. above to $g+f_{n}$ to infer $\int _{E}f\leq {\underline {\lim }}\int _{E}f_{n}$ 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 $|f_{n}|\leq g_{n}$ for each $n\in {\mathbb {Z}}^{+}$.

* $g_{n}\to g$ on E.

* $\lim _{n\to \infty }\int _{E}g_{n}=\int _{E}\lim _{n\to \infty }g_{n}=\int _{E}g$

Now prove that $\lim _{n\to \infty }\int _{E}f_{n}=\int _{E}\lim _{n\to \infty }f_{n}=\int _{E}f$.

The proof should merely reiterate all of the proof of the LDCT, but replacing g by $g_{n}$ where appropriate.