Integration over General Measure Spaces

Let $E \in \mathcal{M}$. Show that a function $f: X \rightarrow \overline{\mathbb {R}}$ is measurable if and only if its restrictions to $E$ and $X \sim E$ are measurable.

Assume that ($X, \mathcal{M}, \mu$) is not complete. Show that there are functions $f: X \rightarrow \mathbb {R}$ and $g: X \rightarrow \mathbb {R}$ for which $f = g$ almost everywhere, and $f$ is measurable but $g$ is not.

Assume that the function $f: X \rightarrow \mathbb {R}$ is measurable. Show that $f^{-1}(E)$ is measurable if $E \subseteq R$ is a Borel set. Is $f^{-1}(E)$ measurable for every Lebesgue measurable set $E \subseteq R$ ?

Show that the function $f: X \rightarrow \overline{\mathbb {R}}$ is measurable if for each rational number $c,\left\{ x \in X \mid f(x) < c\right\}$ is a measurable set.

If $f: \mathbb {R} \rightarrow \mathbb {R}$ is measurable with respect to the $\sigma$-algebra of Lebesgue measurable sets, is it necessarily measurable with respect to the Borel $\sigma$-algebra?

Let $\left\{ f_{n}: X \rightarrow \mathbb {R}\right\}$ be a sequence of measurable functions such that, for each $n, \mu \left\{ x \in X \mid \left|f_{n}(x)-f_{n+1}(x)\right| > 1 / 2^{n}\right\} < 1 / 2^{n}$. Show that $\left\{ f_{n}\right\}$ is pointwise convergent almost everywhere on $X$.

Under the assumptions of Egoroff's Theorem, show that $X = \bigcup_{k = 0}^{\infty } X_{k}$, where each $X_{k}$ is measurable, $\mu \left(X_{0}\right) = 0$ and, for $k \geq 1,\left\{ f_{n}\right\}$ converges uniformly to $f$ on $X_{k}$.

A sequence $\left\{ f_{n}: X \rightarrow \mathbb {R}\right\}$ of measurable functions is said to converge in measure to a measurable function $f: X \rightarrow \mathbb {R}$ provided that for each $\eta > 0$,

Assume that $\mu (X) < \infty$. Show that $\left\{ f_{n}: X \rightarrow \mathbb {R}\right\} \rightarrow f: X \rightarrow \mathbb {R}$ in measure if and only if each subsequence of $\left\{ f_{n}\right\}$ has a further subsequence that converges pointwise almost everywhere on $X$ to $f$. Use this to show that for two sequences that converge in measure, the product sequence also converges in measure to the product of the limits.

Let $\left\{ u_{n}: X \rightarrow [0, \infty ]\right\}$ be a sequence of measurable functions. For $x \in X$, define $f(x) = \sum_{n = 1}^{\infty } u_{n}(x)$. Show that

Show that if $f: X \rightarrow [0, \infty ]$ is measurable, then

Let $g: X \rightarrow [0, \infty ]$ be integrable. Define

Let $f: X \rightarrow [0, \infty ]$ and $g: X \rightarrow [0, \infty ]$ be measurable functions for which $f^{2}$ and $g^{2}$ are integrable. Show that $f \cdot g$ also is integrable.

Let $X$ be the union of a countable, ascending sequence of measurable sets $\left\{ X_{n}\right\}$ and $f: X \rightarrow [0, \infty ]$ be measurable. Show that $f$ is integrable over $X$ if and only if there is an $M \geq 0$ for which $\int_{X_{n}} f d \mu \leq M$ for all $n$.

Let $\eta$ be the counting measure on $2^{\mathrm{N}}$. Characterize the non-negative, real-valued functions (that is, sequences) that are integrable over $\mathbb {N}$ with respect to $\eta$ and the value of $\int_{\mathbb {N}} f d \eta$.

Let $x_{0} \in X$ and $\delta_{x_{0}}$ be the Dirac measure concentrated at $x_{0}$. Characterize the non-negative, real-valued functions on $X$ that are integrable over $X$ with respect to $\delta_{x_{0}}$ and the value of $\int_{X} f d \delta_{x_{0}}$.

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

Let $f:[a, b] \rightarrow \mathbb {R}$ be of bounded variation. Show that there is an absolutely continuous functions $g$ on $[a, b]$ and a function $h$ on $[a, b]$ that is of bounded variation and has $h^{\prime } = 0$ almost everywhere on $[a, b]$, for which $f = g+h$ on $[a, b]$. Then show that this decomposition is unique except for addition of constants.

For the Lebesgue measure space ($\mathbb {R}, \mathcal{M}, m$), let $\nu : \mathcal{M} \rightarrow [0, \infty )$ be a measure and define $h:[a, b] \rightarrow \mathbb {R} h(x) = \nu ([a, x])$ for all $x \in [a, b]$. Show that the function $h$ is absolutely continuous if and only if the measure $\nu$ is absolutely continuous with respect to $m$.

Let $\left\{ \mu_{n}\right\}$ be a sequence of measures on $\mathcal{M}$ for which there is a $c > 0$ such that $\mu_{n}(X) \leq c$ for all $n$. Define $\mu : \mathcal{M} \rightarrow [0, \infty ]$ by

Show that $\mu$ is a measure on $\mathcal{M}$ and each $\mu_{n}$ is absolutely continuous with respect to $\mu$.

Characterize the measure spaces ($X, \mathcal{M}, \mu$) for which the counting measure on $\mathcal{M}$ is absolutely continuous with respect to $\mu$ and those for which, given $x_{0} \in X$, the Dirac measure $\delta_{x_{0}}$ on $\mathcal{M}$ is absolutely continuous with respect to $\mu$.

Let $X = [0,1], \mathcal{M}$ be the collection of Lebesgue measurable subsets of $[0,1], m$ be Lebesgue measure, and $c$ be the counting measure of $\mathcal{M}$. Show that $m$ is absolutely continuous with respect to $c$, but there is no function $f: X \rightarrow [0, \infty ]$ that is integrable over $[0,1]$ with respect to $c$ for which $m(E) = \int_{E} f d c$ for all $E \in \mathcal{M}$.

Let $f: X \rightarrow [0, \infty )$ be integrable. Find the Lebesgue decomposition with respect to $\mu$ of the measure $\nu$ defined by $\nu (E) = \int_{E} f d \mu$ for $E \in \mathcal{M}$.

Let $\mu , \nu$, and $\lambda$ be $\sigma$-finite measures on $\mathcal{M}$.

Let $\mu , \nu_{1}$, and $\nu_{2}$ be measures on $\mathcal{M}$.

Let $\mu$ and $\nu$ be measures on $\mathcal{M}$, define $\lambda = \mu +\nu$, and let $f: X \rightarrow [0, \infty ]$ be measurable. Show that $f$ is integrable with respect to $\lambda$ if and only if it is integrable with respect to both $\mu$ and $\nu$. Also show that if $f$ is integrable with respect to $\lambda$, then

Let $(X, \mathcal{A}, \mu ) = (Y, \mathcal{B}, \nu ) = \left(\mathbb {N}, 2^{\mathbb {N}}, c\right)$. Show that the product measure is the counting measure. State the Fubini and Tonelli Theorems explicitly for this case.

Show that the smallest $\sigma$-algebra that contains all products of Lebesgue measurable subsets of $\mathbb {R}$ is the Borel $\sigma$-algebra $\mathcal{B}\left(\mathbb {R}^{2}\right)$.

Assume $(X, \mathcal{A}, \mu )$ and $(Y, \mathcal{B}, \nu )$ are complete, $h: X \rightarrow [0, \infty ]$ and $g: Y \rightarrow [0, \infty ]$ are nonnegative and integrable, and define $k(x, y) = h(x) \cdot g(y)$. Show that $k$ is a $(\mu \times \nu )$-integrable and

Define $f: \mathbb {N} \times \mathbb {N} \rightarrow \mathbb {R}$ by setting

Show that $f$ is measurable with respect to the product measure space. Also show that

Is this a contradiction either of Fubini's Theorem or Tonelli's Theorem?

Let $(X, \mathcal{A}, \mu )$ be a complete measure space. Consider $\mathbb {N} \times X$ with the product measure $c \times \mu$.

Prove that the conclusion of Tonelli's Theorem holds if one of the spaces is the space ($\mathbb {N}, 2^{\mathbb {N}}, c$) and the other space is a complete measure space that need not be $\sigma$-finite.

If $\left\{ \left(X_{k}, \mathcal{A}_{k}, \mu_{k}\right)\right\}_{k = 1}^{n}$ is a finite collection of measure spaces, define the product measure $\mu_{1} \times \cdots \times \mu_{n}$ on the space $X_{1} \times \cdots \times X_{n}$ by starting with the semi-ring of rectangles of the form $R = A_{1} \times \cdots \times A_{n}$, define $\mu (R) = \prod \mu_{k}\left(A_{k}\right)$, show that $\mu$ is a premeasure and define the product measure to be the Carathéodory extension of $\mu$. Show that if $\left(X_{1} \times \cdots \times X_{p}\right) \times \left(X_{p+1} \times \cdots \times X_{n}\right)$ is identified with ($X_{1} \times \cdots \times X_{n}$), then

A measure space ($X, \mathcal{M}, \mu$) such that $\mu (X) = 1$ is called a probability measure space. Let $\left\{ \left(X_{\lambda }, \mathcal{A}_{\lambda }, \mu_{\lambda }\right)\right\}_{\lambda \in \Lambda }$ be a collection of probability measure spaces parametrized by the set $\Lambda$. Show that it is possible to define a probability measure

on a suitable $\sigma$-algebra on the Cartesian product $\prod_{\lambda \in \Lambda } X_{\lambda }$ so that

when $A = \prod_{\lambda \in \Lambda } A_{\lambda }$. (Note that $\mu (A)$ can only be non-zero if all but a countable number of the $A_{\lambda }$ have $\mu \left(A_{\lambda }\right) = 1$.)

Is every continuous real-valued function $k: \mathbb {R} \times \mathbb {R} \rightarrow \mathbb {R}$ measurable with respect to the product measure space? Show that for any subset $A$ of $\mathbb {R}$, the set $A \times Q$ is measurable with respect to the product measure, irrespective of the measurability of $A$.

Suppose that the function $f: \mathbb {R}^{2} \rightarrow \mathbb {R}$ is continuous with respect to each variable separately, show that it is Lebesgue measurable with respect to $\mu_{2}$.

If $f: \mathbb {R}^{m} \rightarrow \mathbb {R}$ and $g: \mathbb {R}^{k} \rightarrow \mathbb {R}$ are Lebesgue measurable, show that the function $(x, y) \mapsto f(x) \cdot g(y)$ is Lebesgue measurable on $\mathbb {R}^{m+k}$.

Verify that

by checking that, for $(x, y) \neq (0,0)$,

Explain why this example does not contradict either Tonelli's Theorem or Fubini's Theorem.

Show that

Explain why this does not contradict either Tonelli's Theorem or Fubini's Theorem.

Define $f(x, y) = y \exp \left(-\left(1+x^{2}\right) y^{2}\right)$ for $x \geq 0, y \geq 0$. Show that

and conclude from this that

Define $f(x, y) = \exp \left(-x y^{2}\right) \cdot \sin x$ for $x \geq 0, y \geq 0$. Show that for each $b > 0$

and conclude from this that, as an improper Riemann integral,

Assume that $f: \mathbb {R}^{n} \rightarrow [0, \infty )$ is finitely supported and Lebesgue integrable, and that $g: \mathbb {R} \rightarrow \mathbb {R}$ is increasing, bounded, absolutely continuous and $g(0) = 0$. Show that $g \circ f: \mathbb {R}^{n} \rightarrow \mathbb {R}$ is Lebesgue integrable, and that

Show that the product of semi-rings is a semi-ring. Use this to show that the collection of bounded intervals in $\mathbb {R}^{n}$ is a semi-ring.

Let the mapping $\mathcal{N}: \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}$ be one-to-one and onto and there is a $c > 0$ such that

Show that if $f: \mathbb {R}^{n} \rightarrow \mathbb {R}$ is Lebegue measurable, so is $f \circ \mathcal{N}$.