Integration on Product Spaces

Find a monotone class $\mathfrak {M}$ in $\mathbb {R}^{1}$ which is not a $\sigma$-algebra, even though $\mathbb {R}^{1} \in \mathfrak {M}$ and $\mathbb {R}^{1}-A \in \mathfrak {M}$ for every $A \in \mathfrak {M}$.

Suppose $f$ is a Lebesgue measurable nonnegative real function on $\mathbb {R}^{1}$ and $A(f)$ is the ordinate set of $f$. This is the set of all points $(x, y) \in \mathbb {R}^{2}$ for which $0 < y < f(x)$.

Find an example of a positive continuous function $f$ in the open unit square in $\mathbb {R}^{2}$, whose integral (relative to Lebesgue measure) is finite but such that $\varphi (x)$ (in the notation of Theorem 8.8) is infinite for some $x \in (0,1)$.

Suppose $1 \leq p \leq \infty , f \in L^{1}\left(\mathbb {R}^{1}\right)$, and $g \in L^{p}\left(\mathbb {R}^{1}\right)$.

Let $M$ be the Banach space of all complex Borel measures on $\mathbb {R}^{1}$. The norm in $M$ is $\left\| \mu \right\| = \left|\mu \right|\left(\mathbb {R}^{1}\right)$. Associate to each Borel set $E \subset \mathbb {R}^{1}$ the set

If $\mu$ and $\lambda \in M$, define their convolution $\mu * \lambda$ to be the set function given by

for every Borel set $E \subset \mathbb {R}^{1} ; \mu \times \lambda$ is as in Definition 8.7.

(Polar coordinates in $\mathbb {R}^{k}$). Let $S_{k-1}$ be the unit sphere in $\mathbb {R}^{k}$, i.e., the set of all $u \in \mathbb {R}^{k}$ whose distance from the origin 0 is 1. Show that every $x \in \mathbb {R}^{k}$, except for $x = 0$, has a unique representation of the form $x = r u$, where $r$ is a positive real number and $u \in S_{k-1}$. Thus $\mathbb {R}^{k}-\left\{ 0\right\}$ may be regarded as the cartesian product $(0, \infty ) \times S_{k-1}$.

Let $m_{k}$ be Lebesgue measure on $\mathbb {R}^{k}$, and define a measure $\sigma_{k-1}$ on $S_{k-1}$ as follows: If $A \subset S_{k-1}$ and $A$ is a Borel set, let $\tilde{A}$ be the set of all points $r u$, where $0 < r < 1$ and $u \in A$, and define

Prove that the formula

is valid for every nonnegative Borel function $f$ on $\mathbb {R}^{k}$. Check that this coincides with familiar results when $k = 2$ and when $k = 3$.

Suggestion: If $0 < r_{1} < r_{2}$ and if $A$ is an open subset of $S_{k-1}$, let $E$ be the set of all $r u$ with $r_{1} < r < r_{2}, u \in A$, and verify that the formula holds for the characteristic function of $E$. Pass from these to characteristic functions of Borel sets in $\mathbb {R}^{k}$.

Suppose $(X, \mathscr {S}, \mu )$ and $(Y, \mathscr {T}, \lambda )$ are $\sigma$-finite measure spaces, and suppose $\psi$ is a measure on $\mathscr {S} \times \mathscr {T}$ such that

whenever $A \in \mathscr {S}$ and $B \in \mathscr {T}$. Prove that then $\psi (E) = (\mu \times \lambda )(E)$ for every $E \in \mathscr {S} \times \mathscr {T}$.

Suppose $E$ is a dense set in $\mathbb {R}^{1}$ and $f$ is a real function on $\mathbb {R}^{2}$ such that

Suppose $f$ is a real function on $\mathbb {R}^{2}, f_{x}$ is Lebesgue measurable for each $x$, and $f^{y}$ is continuous for each y. Suppose $g: \mathbb {R}^{1} \rightarrow \mathbb {R}^{1}$ is Lebesgue measurable, and put $h(y) = f(g(y), y)$. Prove that $h$ is Lebesgue measurable on $\mathbb {R}^{1}$.

Hint: Define $f_{n}$ as in Exercise 8, put $h_{n}(y) = f_{n}(g(y), y)$, show that each $h_{n}$ is measurable, and that $h_{n}(y) \rightarrow h(y)$.

Let $\mathscr {B}_{k}$ be the $\sigma$-algebra of all Borel sets in $\mathbb {R}^{k}$. Prove that $\mathscr {B}_{m+n} = \mathscr {B}_{m} \times \mathscr {B}_{n}$. This is relevant in Theorem 8.14.

Use Fubini's theorem and the relation

to prove that

If $\mu$ is a complex measure on a $\sigma$-algebra $\mathfrak {M}$, show that every set $E \in \mathfrak {M}$ has a subset $A$ for which

Suggestion: There is a measurable real function $\theta$ so that $d \mu = e^{i \theta } d\left|\mu \right|$. Let $A_{\alpha }$ be the subset of $E$ where $\cos (\theta -\alpha ) > 0$, show that

and integrate with respect to $\alpha$ (as in Lemma 6.3). Show, by an example, that $1 / \pi$ is the best constant in this inequality.

Complete the following proof of Hardy's inequality (Chap. 3, Exercise 14): Suppose $f \geq 0$ on $(0, \infty ), f \in L^{p}, 1 < p < \infty$, and

Write $x F(x) = \int_{0}^{x} f(t) t^{\alpha } t^{-\alpha } d t$, where $0 < \alpha < 1 / q$, use Hölder's inequality to get an upper bound for $F(x)^{p}$, and integrate to obtain

Show that the best choice of $\alpha$ yields

Put $\varphi (t) = 1-\cos t$ if $0 \leq t \leq 2 \pi , \varphi (t) = 0$ for all other real $t$. For $-\infty < x < \infty$, define

Verify the following statements about convolutions of these functions:

(i) $(f * g)(x) = 0$ for all $x$.

(ii) $(g * h)(x) = (\varphi * \varphi )(x) > 0$ on $(0,4 \pi )$.

(iii) Therefore $(f * g) * h = 0$, whereas $f *(g * h)$ is a positive constant.

But convolution is supposedly associative, by Fubini's theorem (Exercise 5(c)). What went wrong?

Prove the following analogue of Minkowski's inequality, for $f \geq 0$ :

Supply the required hypotheses. (Many further developments of this theme may be found in [9]).