Product Measures

Prove that $\mu \times \lambda$, as presented in Definition 8.2.5, is a measure.

Definition 8.2.5: Let $(X, \mathfrak {M}, \mu )$ and $(Y, \mathfrak {N}, \lambda )$ be $\sigma$-finite measure spaces and let $Q \in \mathfrak {M} \otimes \mathfrak {N}$. The product of $\mu$ and $\lambda$ is defined by $(\mu \times \lambda )(Q) = \int_{X} \lambda (Q_{x}) \, d\mu (x) = \int_{Y} \mu (Q^{y}) \, d\lambda (y)$.

Let ( $X, \mathfrak {M}, \mu$ ) and ( $Y, \mathfrak {N}, \lambda$ ) be $\sigma$-finite measure spaces. Prove that $\mu \times \lambda$, as presented in Definition 8.2.5, is $\sigma$-finite.

Definition 8.2.5: Let $(X, \mathfrak {M}, \mu )$ and $(Y, \mathfrak {N}, \lambda )$ be $\sigma$-finite measure spaces and let $Q \in \mathfrak {M} \otimes \mathfrak {N}$. The product of $\mu$ and $\lambda$ is defined by $(\mu \times \lambda )(Q)=\int_{X} \lambda (Q_{x}) \, d\mu (x)=\int_{Y} \mu (Q^{y}) \, d\lambda (y)$, where $Q_{x}=\left\{ y \in Y: (x, y) \in Q\right\}$ and $Q^{y}=\left\{ x \in X: (x, y) \in Q\right\}$.

Let $\mu$ be a counting measure on $\mathbb {P}(\mathbb {N})$. Describe the product measure $\mu \times \mu$.

Let ( $X, \mathfrak {M}, \mu$ ) and ( $Y, \mathfrak {N}, \lambda$ ) be finite measure spaces such that $\mu (X)= \lambda (Y)=1$ and let $E \in \mathfrak {M} \otimes \mathfrak {N}$. Prove that if $(\mu \times \lambda )(E)=1$ then $\lambda \left(E_{x}\right)=1$ for a.e. $x \in X$.

Let $\left(X_{k}, \mathfrak {M}_{k}, \mu_{k}\right)$, for $k=1,2,3$ be $\sigma$-finite measure spaces. Prove that the measures $\left(\mu_{1} \times \mu_{2}\right) \times \mu_{3}$ and $\mu_{1} \times \left(\mu_{2} \times \mu_{3}\right)$ are identical measures on $X_{1} \times X_{2} \times X_{3}$.

Let ( $X, \mathfrak {M}, \mu$ ) and ( $Y, \mathfrak {N}, \lambda$ ) be $\sigma$-finite measure spaces, let $E, F \in \mathfrak {M} \otimes \mathfrak {N}$, and suppose that $\lambda \left(E_{x}\right)=\lambda \left(F_{x}\right)$ for a.e. $[\mu ] x \in X$. Prove that $(\mu \times \lambda )(E)= (\mu \times \lambda )(F)$.

Let ( $X, \mathfrak {M}, \mu$ ) and ( $Y, \mathfrak {N}, \lambda$ ) be $\sigma$-finite measure spaces, let $E \in \mathfrak {M} \otimes \mathfrak {N}$, and suppose that $\lambda \left(E_{x}\right)=0$ for a.e. $[\mu ] x \in X$. Prove that $\mu \left(E^{y}\right)=0$ for a.e. $[\lambda ] y \in Y$.

Let ( $X, \mathfrak {M}, \mu$ ) and ( $Y, \mathfrak {N}, \lambda$ ) be $\sigma$-finite measure spaces, and let $\omega$ be a measure on $\mathfrak {M} \otimes \mathfrak {N}$ such that $\omega (A \times B)=\mu (A) \lambda (B)$ for all $A \in \mathfrak {M}$ and all $B \in \mathfrak {N}$. Prove that $\omega =\mu \times \nu$.

Let ( $X, \mathfrak {M}, \mu$ ) and ( $Y, \mathfrak {N}, \lambda$ ) be finite measure spaces such that $\mu (X)= \lambda (Y)=1$, let $E \subset X \times Y$ such that $(\mu \times \lambda )(E)=\eta^{2}<1$, and let $A=\left\{ x \in X: \lambda \left(E_{x}\right) \geq \eta \right\}$. Find the sharp upper bound for $\mu (A)$.

Let $(X, \mathfrak {M}, \mu )$ and $(Y, \mathfrak {N}, \lambda )$ be finite measure spaces such that $\mu (X)= \lambda (Y)=1$, let $E \subset X \times Y$ such that $\mu \left(E^{y}\right) \geq \alpha$, for all $y \in Y$, let $\beta <\alpha$, and let $A=\left\{ x \in X: \lambda \left(E_{x}\right) \geq \beta \right\}$. Prove that $\mu (A) \geq (\alpha -\beta ) /(1-\beta )$.

Let ( $X, \mathfrak {M}, \mu$ ) and ( $Y, \mathfrak {N}, \lambda$ ) be finite measure spaces such that $\mu (X)= \lambda (Y)=1$, and let $E \in \mathfrak {M} \otimes \mathfrak {N}$ such that $\lambda \left(E_{x}\right) \leq 1 / 2$, for a.e. $[\mu ] x \in X$. Prove that $\lambda \left(\left\{ y \in Y: \mu \left(E^{y}\right)=1\right\} \right) \leq 1 / 2$.

Let $(X, \mathfrak {M}, \mu )$ be a $\sigma$-finite measure space and let $f: X \rightarrow \mathbb {R}$ be a nonnegative, measurable function with graph $G=\left\{ (x, y) \in X \times \mathbb {R}: y=f(x)\right\}$. Prove that $(\mu \times m)(G)=0$.

Let ( $X, \mathfrak {M}, \mu$ ) and ( $Y, \mathfrak {N}, \nu$ ) be finite measure spaces such that $\mu (X)= \nu (Y)=1$, and let $A \in \mathfrak {M} \otimes \mathfrak {N}$. Prove that

Let ( $X, \mathfrak {M}, \mu$ ) be a $\sigma$-finite measure space, let $f: X \rightarrow \mathbb {R}$ be a nonnegative, measurable function, and let $B=\left\{ (x, y) \in X \times \mathbb {R}: 0 \leq y \leq f(x) \text{, if } f(x)<\infty \text{ or } 0 \leq y<\infty \text{ if } f(x)=\infty \right\}$. Prove that $B \in \mathfrak {M} \otimes \mathcal{B}$ and relate $(\mu \times m)(B)$ to the integral of $f$.

Let $\delta$ be the Dirac measure concentrated at $x=0$, and let $E= \left\{ (x, y) \in \mathbb {R}^{2}:\left|x-y\right|<1\right\}$. Find $(\delta \times m)(E)$.

Let $(X, \mathfrak {M}, \mu )$ and $(X, \mathfrak {M}, \lambda )$ be nontrivial $\sigma$-finite measure spaces such that $(\mu \times \lambda )((X \times X) \backslash \Delta )=0$, where $\Delta =\left\{ (x, y) \in X \times X: x=y\right\}$. Prove that there exists $a \in X$ and $\alpha , \beta \in \mathbb {R}$ such that $\mu =\alpha \delta_{a}$ and $\lambda =\beta \delta_{a}$.

The purpose of this exercise is to show that in the absence of $\sigma$ finiteness the product measure need not be unique. Let $\Delta =\left\{ (x, y) \in [0,1]^{2}: x=y\right\}$, let $m$ be the Lebesgue measure on $\mathcal{B}([0,1])$, let $\nu$ be the counting measure on $\mathbb {P}([0,1])$, and let $m \times \nu$ be the Carathéodory extension of the measure $\omega$ defined on measurable rectangles by $\omega (E \times F)=m(E) \nu (F)$. Define $\lambda$ on $\mathcal{B}([0,1]) \otimes \mathbb {P}([0,1])$ by $\lambda (E)=\sup \left\{ \sum_{y \in A} m_{y}(([0,1] \times \left\{ y\right\} ) \cap E): A \text{ is a finite subset of } [0,1]\right\}$, where $m_{y}$ is the Lebesgue measure on $[0,1] \times \left\{ y\right\}$. Prove that:

Let ( $X, \mathfrak {M}, \mu$ ) and ( $Y, \mathfrak {N}, \lambda$ ) be measure spaces and let $E \in \mathfrak {M} \otimes \mathfrak {N}$. By $\mathcal{E}_{\sigma }$ we will denote the collection of countable unions of measurable rectangles and by $\mathcal{E}_{\sigma \delta }$ the collection of countable intersections of sets in $\mathcal{E}_{\sigma }$. Prove that there exists a set $F \in \mathcal{E}_{\sigma \delta }$ such that $E \subset F$ and $(\mu \times \lambda )(E)=(\mu \times \lambda )(F)$.

Let ( $X, \mathfrak {M}, \mu$ ) and ( $X, \mathfrak {M}, \lambda$ ) be $\sigma$-finite measure spaces, and let $E \in \mathfrak {M} \otimes \mathfrak {N}$ be such that $(\mu \times \lambda )(E)=0$. Prove that for $\mu$-a.e. $x \in X$, we have $\lambda \left(E_{x}\right)=0$.

Suppose that ( $X, \mathfrak {M}, \mu$ ) and ( $Y, \mathfrak {N}, \lambda$ ) are $\sigma$-finite measure spaces. Let $A \in \mathfrak {M}$ and $B \in \mathfrak {N}$ satisfy $\mu (A)=0$ and $\lambda (B)=\infty$. Prove that $(\mu \times \lambda )(A \times B)=0$.

Let $E$ be a measurable subset of $[0,1] \times [0,1]$ with the property that it its intersection with every vertical line is at most countable. Prove or disprove: the two-dimensional Lebesgue measure of $E$ is 0 .

Prove that there exists a non-measurable set $E \subset [0,1] \times [0,1]$ with the property that it intersects every vertical line at exactly one point.

Let $f: \mathbb {R}^{2} \rightarrow \mathbb {R}$ be $\mathcal{M} \otimes \mathcal{M}$-measurable, let $A=\left\{ (x, y) \in \mathbb {R}^{2}: f(x, y) \neq 0\right\}$, and suppose that $m \times m(A)=0$. Prove that $m\left(A_{x}\right)=0$ for a.e. $x \in \mathbb {R}$, and that $m\left(A^{y}\right)=0$ for a.e. $y \in \mathbb {R}$.

Let $f$ be a Lebesgue measurable function on $\mathbb {R}$ and let $F(x, y)= f(x-y)$. Prove that $F$ is Lebesgue measurable on $\mathbb {R}^{2}$.

Let $A \subset \mathbb {R}^{p+q}$ and $m_{p+q}(A)=0$. Prove that for almost every $x \in \mathbb {R}^{p}$ the set $A_{x}$ belongs to $\mathcal{M}_{q}$ and $m_{q}\left(A_{x}\right)=0$.

Let ( $X, \mathfrak {M}, \mu$ ) and ( $Y, \mathfrak {N}, \lambda$ ) be finite measure spaces such that $\mu (X)= \lambda (Y)=1$, and let $E \subset X \times Y$ such that $\lambda \left(E_{x}\right)=0$ for a.e. $[\mu ] x \in X$. Suppose that $\lambda \left(\left\{ y \in Y: \mu \left(X \backslash E^{y}\right)<1\right\} \right)>0$. Prove that $E \notin (\mathfrak {M} \otimes \mathfrak {N})$.

Let ( $X, \mathfrak {M}, \mu$ ) be a $\sigma$-finite measure space, let $A \in \mathfrak {M}$, let $0 \leq \alpha \leq \infty$, and define $A_{\alpha }=\left\{ (x, y): x \in A, 0 \leq y \leq \alpha \right\}$ and $A_{\infty }=\left\{ (x, y): x \in A, 0 \leq y<\infty \right\}$. Prove that $A_{\alpha } \in \mathfrak {M} \otimes \mathcal{B}$ and find $\mu \times m\left(A_{\alpha }\right)$.

Let $f$ be a Borel measurable function on $\mathbb {R}$ and $E=\left\{ (x+f(x), x- f(x)): x \in \mathbb {R}\right\}$. Prove or disprove: $m_{2}(E)=0$.

Let $f$ and $g$ be real-valued integrable functions on a measure space $(X, \mathfrak {M}, \mu )$ and define

Prove that

By Example 1.4.13, there is a well-ordered set $W$ and a bijection $j:[0,1] \rightarrow W$ such that for any $x \in [0,1], j(x)$ has at most a countable number of predecessors. Let $Q$ be the set of points $(x, y)$ in $[0,1] \times [0,1]$ such that $j(x)$ precedes $j(y)$, and let $f=\chi_{Q}$. Prove that (8.9) is not true for $(X, \mathfrak {M}, \mu )=(Y, \mathfrak {N}, \lambda )=[0,1]$ with Lebesgue measure. Explain why it does not contradict Theorem 8.2.1.

Theorem 8.2.1, with formulas (8.8) and (8.9): Let $(X, \mathfrak {M}, \mu )$ and $(Y, \mathfrak {N}, \lambda )$ be $\sigma$-finite measure spaces and let $Q \in \mathfrak {M} \otimes \mathfrak {N}$. If $\varphi (x)=\lambda (Q_{x})$ and $\psi (y)=\mu (Q^{y})$ (formula (8.8)), where $Q_{x}=\left\{ y \in Y: (x, y) \in Q\right\}$ and $Q^{y}=\left\{ x \in X: (x, y) \in Q\right\}$, then $\varphi$ is $\mathfrak {M}$-measurable, $\psi$ is $\mathfrak {N}$-measurable and

which is equivalent to $\int_{X} d\mu (x) \int_{Y} \chi_{Q}(x, y) \, d\lambda (y)=\int_{Y} d\lambda (y) \int_{X} \chi_{Q}(x, y) \, d\mu (x)$.

Give an example of a set $Q \subset [0,1] \times [0,1]$ for which both $\varphi (x)$ and $\psi (y)$ (as in (8.8)) exist, but only one of them is measurable.

Formula (8.8): for measure spaces $(X, \mathfrak {M}, \mu )$, $(Y, \mathfrak {N}, \lambda )$, a set $Q \subset X \times Y$ and $x \in X$, $y \in Y$, $\varphi (x)=\lambda (Q_{x})$ and $\psi (y)=\mu (Q^{y})$, where $Q_{x}=\left\{ y \in Y: (x, y) \in Q\right\}$ and $Q^{y}=\left\{ x \in X: (x, y) \in Q\right\}$.

Prove or disprove: there exists a set $Q \subset [0,1] \times [0,1]$ for which $\varphi (x)=1$ and $\psi (y)=0$, where $\varphi$ and $\psi$ are as in (8.8).

Formula (8.8): for measure spaces $(X, \mathfrak {M}, \mu )$, $(Y, \mathfrak {N}, \lambda )$, a set $Q \subset X \times Y$ and $x \in X$, $y \in Y$, $\varphi (x)=\lambda (Q_{x})$ and $\psi (y)=\mu (Q^{y})$, where $Q_{x}=\left\{ y \in Y: (x, y) \in Q\right\}$ and $Q^{y}=\left\{ x \in X: (x, y) \in Q\right\}$.

Let ( $X, \mathfrak {M}, \mu$ ) and ( $Y, \mathfrak {N}, \lambda$ ) be $\sigma$-finite measure spaces such that ( $X \times Y, \mathfrak {M} \otimes \mathfrak {N}, \mu \times \lambda$ ) is complete. Suppose that $B \subset \mathfrak {N}$ and $\lambda (B)=0$. Prove that $\mathfrak {M}=\mathbb {P}(X)$.

Suppose that ( $X, \mathfrak {M}, \mu$ ) and ( $Y, \mathfrak {N}, \lambda$ ) are measure spaces and let $(X, \overline{\mathfrak {M}}, \bar{\mu })$ and $(Y, \overline{\mathfrak {N}}, \bar{\lambda })$ denote their completions. Also, we denote by $(X \times Y, \overline{\mathfrak {M} \otimes \mathfrak {N}}, \overline{\mu \times \lambda }$ ) the completion of ( $X \times Y, \mathfrak {M} \otimes \mathfrak {N}, \mu \times \lambda$ ). Finally, let $(X \times Y, \overline{\overline{\mathfrak {M}} \otimes \overline{\mathfrak {N}}}, \overline{\bar{\mu } \otimes \bar{\lambda }})$ denote the completion of ( $X \times Y, \overline{\mathfrak {M}} \otimes \overline{\mathfrak {N}}, \bar{\mu } \otimes \bar{\lambda }$ ). Prove that

Let $N \subset \mathbb {R}^{p}$ satisfy $m_{p}(N)=0$ and let $B \subset \mathbb {R}^{q}$. Prove that $m_{p+q}(N \times B)=0$.

Prove part (b) of Theorem 8.2.9.

Theorem 8.2.9: Let $p, q, r$ be positive integers satisfying $p+q=r$. Then (a) $\mathcal{B}_{r} \subset \mathcal{M}_{p} \otimes \mathcal{M}_{q} \subset \mathcal{M}_{r}$. (b) $\mathcal{M}_{p} \otimes \mathcal{M}_{q} \neq \mathcal{M}_{r}$. (Part (a) is proved in the text; only part (b) remains.)

Let $p, q, r$ be positive integers satisfying $p+q=r$. Prove that $\mathcal{B}_{p} \otimes \mathcal{B}_{q}= \mathcal{B}_{r}$. Conclude that $\mathcal{B}_{r} \neq \mathcal{M}_{p} \otimes \mathcal{M}_{q}$.