Product Measures

Suppose $(X, \mathcal{S})$ and $(Y, \mathcal{T})$ are measurable spaces. Prove that if $A$ is a nonempty subset of $X$ and $B$ is a nonempty subset of $Y$ such that $A \times B \in \mathcal{S} \otimes \mathcal{T}$, then $A \in \mathcal{S}$ and $B \in \mathcal{T}$.

Suppose $(X, \mathcal{S})$ is a measurable space. Prove that if $E \in \mathcal{S} \otimes \mathcal{S}$, then

Let $\mathcal{B}$ denote the $\sigma$-algebra of Borel subsets of $\mathbb {R}$. Show that there exists a set $E \subseteq \mathbb {R} \times \mathbb {R}$ such that $[E]_{a} \in \mathcal{B}$ and $[E]^{a} \in \mathcal{B}$ for every $a \in \mathbb {R}$, but $E \notin \mathcal{B} \otimes \mathcal{B}$.

Suppose $(X, \mathcal{S})$ and $(Y, \mathcal{T})$ are measurable spaces. Prove that if $f: X \rightarrow \mathbb {R}$ is $\mathcal{S}$-measurable and $g: Y \rightarrow \mathbb {R}$ is $\mathcal{T}$-measurable and $h: X \times Y \rightarrow \mathbb {R}$ is defined by $h(x, y) = f(x) g(y)$, then $h$ is $(\mathcal{S} \otimes \mathcal{T})$-measurable.

Verify the assertion in Example 5.11 that the collection of finite unions of intervals of $\mathbb {R}$ is closed under complementation.

Verify the assertion in Example 5.12 that the collection of countable unions of intervals of $\mathbb {R}$ is not closed under complementation.

Suppose $\mathcal{A}$ is a nonempty collection of subsets of a set $W$. Show that $\mathcal{A}$ is an algebra on $W$ if and only if $\mathcal{A}$ is closed under finite intersections and under complementation.

Suppose $\mu$ is a measure on a measurable space $(X, \mathcal{S})$. Prove that the following are equivalent:

1. The measure $\mu$ is $\sigma$-finite.

2. There exists an increasing sequence $X_{1} \subseteq X_{2} \subseteq \cdots$ of sets in $\mathcal{S}$ such that $X = \bigcup_{k = 1}^{\infty } X_{k}$ and $\mu \left(X_{k}\right) < \infty$ for every $k \in \mathbb {N}$.

3. There exists a disjoint sequence $X_{1}, X_{2}, X_{3}, \ldots$ of sets in $\mathcal{S}$ such that $X = \bigcup_{k = 1}^{\infty } X_{k}$ and $\mu \left(X_{k}\right) < \infty$ for every $k \in \mathbb {N}$.

Suppose $\mu$ and $\nu$ are $\sigma$-finite measures. Prove that $\mu \times \nu$ is a $\sigma$-finite measure.

Suppose $(X, \mathcal{S}, \mu )$ and $(Y, \mathcal{T}, \nu )$ are $\sigma$-finite measure spaces. Prove that if $\omega$ is a measure on $\mathcal{S} \otimes \mathcal{T}$ such that $\omega (A \times B) = \mu (A) \nu (B)$ for all $A \in \mathcal{S}$ and all $B \in \mathcal{T}$, then $\omega = \mu \times \nu$.

Suppose $(X, \mathcal{S})$ is a measurable space and $f: X \rightarrow [0, \infty ]$ is a function. Let $\mathcal{B}$ denote the $\sigma$-algebra of Borel subsets of $(0, \infty )$. Prove that $U_{f} \in \mathcal{S} \otimes \mathcal{B}$ if and only if $f$ is an $\mathcal{S}$-measurable function.

Suppose $(X, \mathcal{S})$ is a measurable space and $f: X \rightarrow \mathbb {R}$ is a function. Let $\operatorname {graph}(f) \subseteq X \times \mathbb {R}$ denote the graph of $f$ :

Let $\mathcal{B}$ denote the $\sigma$-algebra of Borel subsets of $\mathbb {R}$. Prove that graph $(f) \in \mathcal{S} \otimes \mathcal{B}$ if $f$ is an $\mathcal{S}$-measurable function.

Show that a set $G \subseteq \mathbb {R}^{n}$ is open in $\mathbb {R}^{n}$ if and only if for each $\left(b_{1}, \ldots , b_{n}\right) \in G$, there exists $r > 0$ such that

Show that there exists a set $E \subseteq \mathbb {R}^{2}$ (thinking of $\mathbb {R}^{2}$ as equal to $\mathbb {R} \times \mathbb {R}$) such that the cross sections $[E]_{a}$ and $[E]^{a}$ are open subsets of $\mathbb {R}$ for every $a \in \mathbb {R}$, but $E \notin \mathcal{B}_{2}$.

Suppose $(X, \mathcal{S}),(Y, \mathcal{T})$, and $(Z, \mathcal{U})$ are measurable spaces. We can define $\mathcal{S} \otimes \mathcal{T} \otimes \mathcal{U}$ to be the smallest $\sigma$-algebra on $X \times Y \times Z$ that contains

Prove that if we make the obvious identifications of the products $(X \times Y) \times Z$ and $X \times (Y \times Z)$ with $X \times Y \times Z$, then

Show that Lebesgue measure on $\mathbb {R}^{n}$ is translation invariant. More precisely, show that if $E \in \mathcal{B}_{n}$ and $a \in \mathbb {R}^{n}$, then $a+E \in \mathcal{B}_{n}$ and $\lambda_{n}(a+E) = \lambda_{n}(E)$, where

Suppose $f: \mathbb {R}^{n} \rightarrow \mathbb {R}$ is $\mathcal{B}_{n}$-measurable and $t \in \mathbb {R} \backslash \left\{ 0\right\}$. Define $f_{t}: \mathbb {R}^{n} \rightarrow \mathbb {R}$ by $f_{t}(x) = f(t x)$.

Suppose $\lambda$ denotes Lebesgue measure on $(\mathbb {R}, \mathcal{L})$, where $\mathcal{L}$ is the $\sigma$-algebra of Lebesgue measurable subsets of $\mathbb {R}$. Show that there exist subsets $E$ and $F$ of $\mathbb {R}^{2}$ such that

• $F \in \mathcal{L} \otimes \mathcal{L}$ and $(\lambda \times \lambda )(F) = 0$;

• $E \subseteq F$ but $E \notin \mathcal{L} \otimes \mathcal{L}$.

Suppose $m \in \mathbb {N}$. Verify that the collection of sets $\mathcal{E}_{m}$ that appears in the proof of 5.41 is a monotone class.

Show that the open unit ball in $\mathbb {R}^{n}$ is an open subset of $\mathbb {R}^{n}$.

Suppose $G_{1}$ is a nonempty subset of $\mathbb {R}^{m}$ and $G_{2}$ is a nonempty subset of $\mathbb {R}^{n}$. Prove that $G_{1} \times G_{2}$ is an open subset of $\mathbb {R}^{m} \times \mathbb {R}^{n}$ if and only if $G_{1}$ is an open subset of $\mathbb {R}^{m}$ and $G_{2}$ is an open subset of $\mathbb {R}^{n}$.

Suppose $F_{1}$ is a nonempty subset of $\mathbb {R}^{m}$ and $F_{2}$ is a nonempty subset of $\mathbb {R}^{n}$. Prove that $F_{1} \times F_{2}$ is a closed subset of $\mathbb {R}^{m} \times \mathbb {R}^{n}$ if and only if $F_{1}$ is a closed subset of $\mathbb {R}^{m}$ and $F_{2}$ is a closed subset of $\mathbb {R}^{n}$.

Suppose $E$ is a subset of $\mathbb {R}^{m} \times \mathbb {R}^{n}$ and

For readers familiar with the gamma function $\Gamma$ : Prove that

for every positive integer $n$.

Define $f: \mathbb {R}^{2} \rightarrow \mathbb {R}$ by