Math Problems Vault

Problem 11.49

Is every continuous real-valued function $k: \mathbb {R} \times \mathbb {R} \rightarrow \mathbb {R}$ measurable with respect to the product measure space? Show that for any subset $A$ of $\mathbb {R}$ , the set $A \times Q$ is measurable with respect to the product measure, irrespective of the measurability of $A$ .

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Problem 11.50

Suppose that the function $f: \mathbb {R}^{2} \rightarrow \mathbb {R}$ is continuous with respect to each variable separately, show that it is Lebesgue measurable with respect to $\mu_{2}$ .

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Problem 11.51

If $f: \mathbb {R}^{m} \rightarrow \mathbb {R}$ and $g: \mathbb {R}^{k} \rightarrow \mathbb {R}$ are Lebesgue measurable, show that the function $(x, y) \mapsto f(x) \cdot g(y)$ is Lebesgue measurable on $\mathbb {R}^{m+k}$ .

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Problem 11.52

Verify that

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

by checking that, for $(x, y) \neq (0,0)$ ,

\frac{\partial }{\partial y}\left[\frac{y}{x^{2}+y^{2}}\right] = \frac{x^{2}-y^{2}}{\left(x^{2}+y^{2}\right)^{2}} = \frac{\partial }{\partial x}\left[\frac{-x}{x^{2}+y^{2}}\right]

Explain why this example does not contradict either Tonelli's Theorem or Fubini's Theorem.

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Problem 11.53

Show that

\int _{-1}^{1}\left[\int _{-1}^{1} \frac{x y}{\left(x^{2}+y^{2}\right)^{2}} d x\right] d y \neq \int _{-1}^{1}\left[\int _{-1}^{1} \frac{x y}{\left(x^{2}+y^{2}\right)^{2}} d y\right] d x

Explain why this does not contradict either Tonelli's Theorem or Fubini's Theorem.

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Problem 11.54

Define $f(x, y) = y \exp \left(-\left(1+x^{2}\right) y^{2}\right)$ for $x \geq 0, y \geq 0$ . Show that

\int _{0}^{\infty }\left[\int _{0}^{\infty } f(x, y) d y\right] d x = \int _{0}^{\infty }\left[\int _{0}^{\infty } f(x, y) d x\right] d y

and conclude from this that

\int _{0}^{\infty } \exp \left(-x^{2}\right) d x = \sqrt{\pi } / 2

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Problem 11.55

Define $f(x, y) = \exp \left(-x y^{2}\right) \cdot \sin x$ for $x \geq 0, y \geq 0$ . Show that for each $b > 0$

\int _{0}^{b}\left[\int _{0}^{\infty } f(x, y) d y\right] d x = \int _{0}^{\infty }\left[\int _{0}^{b} f(x, y) d x\right] d y

and conclude from this that, as an improper Riemann integral,

\int _{0}^{\infty } \frac{\sin x}{x^{1 / 2}} d x \equiv \lim _{b \rightarrow \infty } \int _{0}^{b} \frac{\sin x}{x^{1 / 2}} d x = \sqrt{\pi / 2}

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Problem 11.56

Assume that $f: \mathbb {R}^{n} \rightarrow [0, \infty )$ is finitely supported and Lebesgue integrable, and that $g: \mathbb {R} \rightarrow \mathbb {R}$ is increasing, bounded, absolutely continuous and $g(0) = 0$ . Show that $g \circ f: \mathbb {R}^{n} \rightarrow \mathbb {R}$ is Lebesgue integrable, and that

\int _{\mathbb {R}^{n}} g(f(x)) d \mu _{n}(x) = \int _{0}^{\infty }\left[\mu _{n}\left\{ x \in \mathbb {R}^{n} \mid 0 \leq t \leq f(x)\right\} \cdot g^{\prime }(t)\right] d t

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Problem 11.57

Show that the product of semi-rings is a semi-ring. Use this to show that the collection of bounded intervals in $\mathbb {R}^{n}$ is a semi-ring.

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Problem 11.58

Let the mapping $\mathcal{N}: \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}$ be one-to-one and onto and there is a $c > 0$ such that

\left\| \mathcal{N}(x)-\mathcal{N}(y)\right\| \geq c \cdot \left\| x-y\right\| \forall x, y \in \mathbb {R}^{n}

Show that if $f: \mathbb {R}^{n} \rightarrow \mathbb {R}$ is Lebegue measurable, so is $f \circ \mathcal{N}$ .

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