Math Problems Vault

Problem 5B.1

(a)(b)

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(a)

Let $\lambda$ denote Lebesgue measure on $[0,1]$ . Show that

\int _{[0,1]} \int _{[0,1]} \frac{x^{2}-y^{2}}{\left(x^{2}+y^{2}\right)^{2}} d \lambda (y) d \lambda (x) = \frac{\pi }{4}

and

\int _{[0,1]} \int _{[0,1]} \frac{x^{2}-y^{2}}{\left(x^{2}+y^{2}\right)^{2}} d \lambda (x) d \lambda (y) = -\frac{\pi }{4}

(b)

Explain why (a) violates neither Tonelli's Theorem nor Fubini's Theorem.

Problem 5B.2

(a)(b)

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(a)

Give an example of a doubly indexed collection $\left\{ x_{m, n}: m, n \in \mathbb {N}\right\}$ of real numbers such that

\sum _{m = 1}^{\infty } \sum _{n = 1}^{\infty } x_{m, n} = 0 \quad \text{ and } \quad \sum _{n = 1}^{\infty } \sum _{m = 1}^{\infty } x_{m, n} = \infty

(b)

Explain why (a) violates neither Tonelli's Theorem nor Fubini's Theorem.

Problem 5B.3

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.

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Remark.

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Definition (region under the graph; $U_{f}$ ): Suppose $X$ is a set and $f: X \rightarrow [0, \infty ]$ is a function. Then the region under the graph of $f$ , denoted $U_{f}$ , is defined by $U_{f} = \left\{ (x, t) \in X \times (0, \infty ): 0 < t < f(x)\right\}$ .

Problem 5B.4

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 $f$ :

\operatorname {graph}(f) = \left\{ (x, f(x)): x \in X\right\}

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.

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