Products of Measure Spaces

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 $\omega = \mu \times \nu$.