Math Problems Vault

Problem 11.41

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.

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Problem 11.42

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

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Problem 11.43

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

\int _{X \times Y} k d(\mu \times \nu ) = \int _{X} h d \mu \cdot \int _{Y} g d \nu

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Problem 11.44

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

f(m, n) = \begin{cases} 2-2^{-m} & \text{if } m = n \\ -2+2^{-m} & \text{if } m = n+1 \\ 0 & \text{otherwise} \end{cases}

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

\int _{\mathbb {N}}\left[\int _{\mathbb {N}} f(m, n) d c(m)\right] d c(n) \neq \int _{\mathbb {N}}\left[\int _{\mathbb {N}} f(m, n) d c(n)\right] d c(m)

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

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Problem 11.45

(i)(ii)(iii)(iv)

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

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(i)

Show that a subset $E$ of $\mathbb {N} \times X$ is measurable with respect to the product measure space if and only if for each $k, E_{k} = \left\{ x \in X \mid (k, x) \in E\right\}$ is measurable with respect to $\mu$ .

(ii)

Show that a function $f: \mathbb {N} \times X \rightarrow \mathbb {R}$ is measurable with respect to the product measure space if and only if for each $k, f(k, \cdot ): X \rightarrow \mathbb {R}$ is measurable with respect to $\mu$ .

(iii)

Show that a function $f: \mathbb {N} \times X \rightarrow \mathbb {R}$ is integrable over $\mathbb {N} \times X$ with respect to $c \times \mu$ if and only if for each $k, f(k, \cdot ): X \rightarrow \mathbb {R}$ is integrable over $X$ with respect to $\mu$ and

\sum _{k = 1}^{\infty } \int _{X}\left|f(k, x)\right| d \mu (x) < \infty

(iv)

Show that if the function $f: \mathbb {N} \times X \rightarrow \mathbb {R}$ is integrable over $\mathbb {N} \times X$ with respect to $c \times \mu$ , then

\int _{\mathbb {N} \times X} f d(c \times \mu ) = \sum _{k = 1}^{\infty } \int _{X} f(k, x) d \mu (x) < \infty

Problem 11.46

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.

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Problem 11.47

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

\left(\mu _{1} \times \cdots \times \mu _{p}\right) \times \left(\mu _{p+1} \times \cdots \times \mu _{n}\right) = \mu _{1} \times \cdots \times \mu _{n} .

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Problem 11.48

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

\mu = \prod _{\lambda \in \Lambda } \mu _{\lambda }

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

\mu (A) = \prod _{\lambda \in \Lambda } \mu _{\lambda }\left(A_{\lambda }\right)

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 $\mu \left(A_{\lambda }\right) = 1$ .)

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