Measurability on Cartesian Products

Prove the second equality in (8.1).

Formula (8.1): For $E \subset X \times Y$ and $x \in X$, where $E_{x}=\left\{ y \in Y:(x, y) \in E\right\}$ denotes the $x$-section of $E$,

Prove the second equality in (8.2).

Formula (8.2): For $\left\{ E_{k}\right\}_{k=1}^{\infty }$ subsets of $X \times Y$, $E=\cup_{k=1}^{\infty } E_{k}$ and $x \in X$, where $E_{x}=\left\{ y \in Y:(x, y) \in E\right\}$ denotes the $x$-section of $E$,

Let $(X, \mathfrak {M})$ and $(Y, \mathfrak {N})$ be measurable spaces. Prove or disprove: $A \times B \in \mathfrak {M} \otimes \mathfrak {N}$ if and only if $A \in \mathfrak {M}$ and $B \in \mathfrak {N}$.

Let $\mathcal{B}$ denote the $\sigma$-algebra of Borel subsets of $\mathbb {R}$. Prove that there exists a set $E \subset \mathbb {R} \times \mathbb {R}$ such that $E_{x} \in \mathcal{B}$ and $E^{y} \in \mathcal{B}$ for every $x, y \in \mathbb {R}$, but $E \notin \mathcal{B} \otimes \mathcal{B}$.

Prove the implication in (8.4).

Definition 8.1.6: A set $Q \subset X \times Y$ is an elementary set if it is a finite disjoint union of measurable rectangles $A \times B$, $A \in \mathfrak {M}$, $B \in \mathfrak {N}$. We denote by $\mathcal{E}$ the collection of elementary sets.

Formulas (8.3)-(8.4): For measurable rectangles,

Equalities in (8.3) imply that

Prove or disprove: If $A, B$ are Lebesgue measurable subsets of $\mathbb {R}^{2}$ then so is $A+B$. Here $A+B=\left\{ x+y: x \in A, y \in B\right\}$.

Suppose ( $X, \mathfrak {M}$ ) is a measurable space. Prove that if $E \in \mathfrak {M} \otimes \mathfrak {M}$, then $\left\{ x \in X:(x, x) \in E\right\} \in \mathfrak {M}$.

Let ( $X, \mathfrak {M}$ ) and ( $Y, \mathfrak {N}$ ) be measurable spaces and let $\mathcal{A}=\left\{ E \subset Y: X \times E \in \mathfrak {M} \otimes \mathfrak {N}\right\}$. Prove that $\mathcal{A}$ is a $\sigma$-algebra that contains $\mathfrak {N}$.

Let $\left(X_{k}, \mathfrak {M}_{k}\right)$, for $k=1,2,3$ be measurable spaces. Prove that $\mathfrak {M}_{1} \otimes \left(\mathfrak {M}_{2} \otimes \mathfrak {M}_{3}\right)=\left(\mathfrak {M}_{1} \otimes \mathfrak {M}_{2}\right) \otimes \mathfrak {M}_{3}$.

Let $X, Y$ be nonempty sets and let $\mathcal{A} \subset \mathbb {P}(X)$. Prove that $\sigma (\mathcal{A}) \times Y= \sigma (\mathcal{A} \times Y)$.

Let $A$ be a Borel set in $\mathbb {R}^{p}$. Prove that $A \times \mathbb {R}^{q}$ is a Borel set in $\mathbb {R}^{p+q}$.

Let $\mathcal{G}$ be a monotone class in $X \times Y$, let $P \subset X \times Y$, and

Prove that:

Let $\mathcal{G}$ be a monotone class that is closed under finite disjoint unions. Prove that it is closed under countable disjoint unions.

Give an example of a monotone class $\mathcal{G}$ on $\mathbb {R}$ that satisfies

1. $\mathbb {R}$ belongs to $\mathcal{G}$, and

2. if $A \in \mathcal{G}$ then its complement $A^{c}$ is in $\mathcal{G}$, but it is not a $\sigma$-algebra.

Let $\mathcal{G}$ be a monotone class and let $\mathcal{F}=\left\{ A \in \mathcal{G}: A^{C} \in \mathcal{G}\right\}$. Prove that $\mathcal{F}$ is a monotone class.

Let $\mathcal{G}$ be a monotone class on $X$, let $\mathcal{C}$ be a collection of subsets of $X$, and let $\mathcal{A}=\left\{ A \in \mathcal{G}: A \cap C \in \mathcal{G} \text{, for all } C \in \mathcal{C}\right\}$. Prove that $\mathcal{A}$ is a monotone class.

Let $\mathcal{G}$ be a monotone class, let $\mathcal{S} \subset \mathcal{G}$ be an algebra, and let $\mathcal{F}=\left\{ A \in \mathcal{G}: A \cup B \in \mathcal{G} \text{, for all } B \in \mathcal{S}\right\}$. Prove that $\mathcal{F}$ is a monotone class.

Let $\mathcal{G}$ be a monotone class on $X$ that is also an algebra. Prove that $\mathcal{G}$ is a $\sigma$-algebra.

Let $\mathcal{S}$ be an algebra, let $\mathcal{G}$ be the smallest monotone class that contains $\mathcal{S}$, and let $\mathcal{T}=\left\{ A \in \mathcal{G}: A \cup B \in \mathcal{G} \text{, for all } B \in \mathcal{G}\right\}$. Prove that $\mathcal{T}$ is a monotone class that contains $\mathcal{S}$.

Let $\mathcal{A}$ be an algebra on $X$. Prove that the smallest monotone class that contains $\mathcal{A}$ is the smallest $\sigma$-algebra that contains $\mathcal{A}$.

${ }^{*}$ Let $\mathcal{A}$ be an algebra on $X$, let $\sigma (\mathcal{A})$ be the $\sigma$-algebra generated by $\mathcal{A}$, and let $\mu , \nu$ be finite measures on $X$ that agree on $\mathcal{A}$.

Let $X$ be a disjoint union $X=\cup_{i=1}^{\infty } X_{i}$, and for each $i \in \mathbb {N}$ let $\mathcal{G}_{i}$ be a monotone class on $X_{i}$. Prove or disprove: $\cup_{i=1}^{\infty } \mathcal{G}_{i}$ is a monotone class.

Let $\left\{ \mathcal{G}_{i}\right\}_{i=1}^{\infty }$ be nested monotone classes on $X$, i.e., $\mathcal{G}_{i} \subset \mathcal{G}_{i+1}$ for all $i \in \mathbb {N}$. Prove or disprove: $\cup_{i=1}^{\infty } \mathcal{G}_{i}$ is a monotone class.

Prove assertion (b) of Theorem 8.1.17.

Theorem 8.1.17: Let $f$ be a $(\mathfrak {M} \otimes \mathfrak {N})$-measurable function on $X \times Y$. Then: (a) For every $x \in X$, $f_{x}$ is a $\mathfrak {N}$-measurable function on $Y$. (b) For every $y \in Y$, $f^{y}$ is a $\mathfrak {M}$-measurable function on $X$. (Part (a) is proved in the text; only part (b) remains.)

Let $(X, \mathfrak {M})$ and $(Y, \mathfrak {N})$ be measurable spaces, let $f: X \rightarrow \mathbb {R}$ be $\mathfrak {M}$ measurable, let $g: Y \rightarrow \mathbb {R}$ be $\mathfrak {N}$-measurable, and let $h: X \times Y \rightarrow \mathbb {R}$ be defined by $h(x, y)=f(x) g(y)$. Prove that $h$ is $\mathfrak {M} \otimes \mathfrak {N}$-measurable.

Let ( $X, \mathfrak {M}$ ) be a measurable space, and let let $f: X \rightarrow \mathbb {R}$ be a nonnegative function. Prove that $f$ is measurable if and only if the set $\left\{ (x, y) \in X \times \mathbb {R}: 0 \leq y \leq f(x)\right\}$ is measurable, meaning that it belongs to $\mathfrak {M} \otimes \mathcal{M}$.

Let $f:[a, b] \times \mathbb {R} \rightarrow \mathbb {R}$ such that $f_{x}$ is continuous for every $x \in [a, b]$ and $f^{y}$ is Lebesgue measurable for every $y \in \mathbb {R}$. Prove that $f$ is $\mathcal{B}([a, b]) \otimes \mathcal{M}$ measurable.

Let $(X, \mathfrak {M})$ and $(Y, \mathfrak {N})$ be measurable spaces, let $f: X \rightarrow Y$, and suppose that the graph $f$ is $\mathfrak {M} \otimes \mathfrak {N}$-measurable. Prove or disprove: $f$ is measurable, meaning that if $A \in \mathfrak {N}$ then $f^{-1}(A) \in \mathfrak {M}$.

Let ( $X, \mathfrak {M}$ ) and ( $Y, \mathfrak {N}$ ) be measurable spaces and let $f: X \rightarrow Y$ be measurable, meaning that if $A \in \mathfrak {N}$ then $f^{-1}(A) \in \mathfrak {M}$. Prove or disprove: the graph $f$ is $\mathfrak {M} \otimes \mathfrak {N}$-measurable.

Let $(X, \mathfrak {M})$ and $(Y, \mathfrak {N})$ be measurable spaces. Prove that $\mathfrak {M} \otimes \mathfrak {N}$ is the smallest $\sigma$-algebra that makes projections measurable. (Projections $P_{1}$ and $P_{2}$ are defined by $P_{1}(x, y)=x$ and $P_{2}(x, y)=y$.)

Let ( $X, \mathfrak {M}$ ) be a measurable space and let $f: X \times [0,1] \rightarrow \mathbb {R}$ be a function such that $f^{y}$ is measurable for each $y \in [0,1]$ and $f_{x}$ is continuous for each $x \in X$. Show that the set $E_{\varepsilon , \delta }:=\left\{ x: \left|f(x, y)-f(x, 0)\right| \geq \varepsilon \text{, for all } y< \delta \right\}$ is measurable.

Let $f: \mathbb {R}^{2} \rightarrow \mathbb {R}$ be a function for which both $f_{x}$ and $f^{y}$ are Borel measurable. Prove or disprove: $f$ is Borel measurable.

Let $X$ and $Y$ be sets, let $E \subset X \times Y$, and let $f=\chi_{E}$. Prove that $f_{x}=\chi_{E_{x}}$ and $f^{y}=\chi_{E^{y}}$.