Integration of General Measurable Functions: The Dominated Convergence Theorem and the Vitali Convergence Theorem

Let $\mathcal{M}$ be the $\sigma$-algebra of all subsets of $\mathbb {N}$ and $\eta$ be the counting measure of $\mathcal{M}$. Let $\mathcal{F}$ be a uniformly pointwise bounded collection of functions that are integrable over $\mathbb {N}$ with respect to $\eta$

Show that if the function $f: X \rightarrow \mathbb {R}$ is integrable and $E \in \mathcal{M}$, then $f: E \rightarrow \mathbb {R}$ is integrable.

Assume that $\mu (X) = 0$ and the function $f: X \rightarrow \mathbb {R}$ take the constant value $\infty$. Show that $\int_{X} f d \mu = 0$.

Provide an example of a function $f: X \rightarrow \mathbb {R}$ for which $\left|f\right|: X \rightarrow \mathbb {R}$ is integrable but $f: X \rightarrow \mathbb {R}$ is not integrable.

Let the function $f: X \rightarrow \mathbb {R}$ be finitely supported, bounded, and measurable. Show that

where $\psi$ and $\varphi$ range over all simple functions on $X$ for which $\psi \leq f \leq \varphi$ on $X$.

Assume that ($X, \mathcal{M},, \mu$) is complete, the function $f: X \rightarrow \mathbb {R}$ is finitely supported and bounded, and

where $\psi$ and $\varphi$ range over all simple functions on $X$ for which $\psi \leq f \leq \varphi$ on $X$. Prove that $f$ is measurable.

Assume that ($X, \mathcal{M}, \mu$) has the property that for each $\epsilon > 0, X$ is the union of a finite collection of measurable sets, each of measure at most $\epsilon$. Let the sequence $\left\{ f_{n}: X \rightarrow [0, \infty ]\right\}$ be uniformly integrable and converge pointwise on $X$ to $f: X \rightarrow [0, \infty ]$. Show that the function $f$ is integrable.

On a set $X$, consider $\sigma$-algebra $\mathcal{M} = \left\{ \emptyset , X\right\}$, and define $\mu (\emptyset ) = 0, \mu (X) = 1$. Show that every collection of integrable functions on $X$ is uniformly integrable and tight. Find a sequence of simple functions that converges pointwise to a non-integrable function.

Prove Corollary 15.

Corollary 15: Let $\left\{ f_{n}: X \rightarrow [0, \infty )\right\}$ be a sequence of non-negative, integrable functions. Suppose that $\left\{ f_{n}(x)\right\} \rightarrow 0$ for all $x \in X$. Then $\lim_{n \rightarrow \infty } \int_{X} f_{n} \, d\mu = 0$ if and only if $\left\{ f_{n}\right\}$ is uniformly integrable and tight.

Deduce the Dominated Convergence Theorem from the Vitali Convergence Theorem.

Show that a collection $\mathcal{F}$ of integrable functions over $X$ is uniformly integrable if and only if for each $\epsilon > 0$, there is a $\delta > 0$ such that for any $E \in \mathcal{M}$ and any $f \in \mathcal{F}$,

Let $\eta$ be another measure on $\mathcal{M}$. Assume that the function $f: X \rightarrow [0, \infty ]$ is measurable. Is

Let $\mathcal{M}_{0}$ be a $\sigma$-algebra that is contained in $\mathcal{M}, \mu_{0}$ be the restriction of $\mu$ to $\mathcal{M}_{0}$, and the function $f: X \rightarrow [0, \infty ]$ be measurable with respect to $\mathcal{M}_{0}$. Show that $f$ is measurable with respect to $\mathcal{M}$ and

Provide an example for which this inequality is strict.

Let $\nu$ be a signed measure on a $\sigma$-algebra of subsets of $X$ that has Jordan decomposition $\nu = \nu^{+}-\nu^{-}$. For $f: X \rightarrow \mathbb {R}$, define

provided that $f$ is integrable over $X$ with respect to both $\nu^{+}$and $\nu^{-}$. Show that if $\left|f\right| \leq M$ on $X$, then

Moreover, if $\left|\nu \right|(X) < \infty$, show that there is a measurable function $f$ with $\left|f\right| \leq 1$ on $X$ for which