Chapter 9

Real and Complex Measures

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Problem 9A.1

Prove or give a counterexample: If $\nu$ is a real measure on a measurable space $(X, \mathcal{S})$ and $A, B \in \mathcal{S}$ are such that $\nu (A) \geq 0$ and $\nu (B) \geq 0$ , then $\nu (A \cup B) \geq 0$ .

Problem 9A.2

Suppose $\nu$ is a real measure on $(X, \mathcal{S})$ . Define $\mu : \mathcal{S} \rightarrow [0, \infty )$ by

\mu (E) = \left|\nu (E)\right|.

Prove that $\mu$ is a (positive) measure on $(X, \mathcal{S})$ if and only if the range of $\nu$ is contained in $[0, \infty )$ or the range of $\nu$ is contained in $(-\infty , 0]$ .

Problem 9A.3

Suppose $\nu$ is a complex measure on a measurable space $(X, \mathcal{S})$ . Prove that $\left|\nu \right|(X) = \nu (X)$ if and only if $\nu$ is a (positive) measure.

Problem 9A.4

Suppose $\nu$ is a complex measure on a measurable space $(X, \mathcal{S})$ . Prove that if $E \in \mathcal{S}$ then

\left|\nu \right|(E) = \sup \left\{ \sum _{k = 1}^{\infty }\left|\nu \left(E_{k}\right)\right|: E_{1}, E_{2}, \ldots \text{ is a disjoint sequence in } \mathcal{S} \text{ such that } E = \bigcup _{k = 1}^{\infty } E_{k}\right\}

Problem 9A.5

Suppose $\mu$ is a (positive) measure on a measurable space $(X, \mathcal{S})$ and $h$ is a nonnegative function in $\mathcal{L}^{1}(\mu )$ . Let $\nu$ be the (positive) measure on $(X, \mathcal{S})$ defined by $d \nu = h d \mu$ . Prove that

\int f \; d \nu = \int f h \; d \mu

for all $S$ -measurable functions $f: X \rightarrow [0, \infty ]$ .

Problem 9A.6

Suppose $(X, \mathcal{S}, \mu )$ is a (positive) measure space. Prove that

\operatorname {AC}_{\mu } = \left\{ h \; d \mu : h \in \mathcal{L}^{1}(\mu )\right\}

is a closed subspace of $\mathcal{M}_{\mathrm{F}}(\mathcal{S})$ .

Problem 9A.7

(a)(b)

(a)

Suppose $\mathcal{B}$ is the collection of Borel subsets of $\mathbb {R}$ . Show that the Banach space $\mathcal{M}_{\mathbf{F}}(\mathcal{B})$ is not separable.

(b)

Give an example of a measurable space $(X, \mathcal{S})$ such that the Banach space $\mathcal{M}_{\mathbf{F}}(\mathcal{S})$ is infinite-dimensional and separable.

Problem 9A.8

(a)(b)

Suppose $t > 0$ and $\lambda$ is Lebesgue measure on the $\sigma$ -algebra of Borel subsets of $[0, t]$ . Suppose $h:[0, t] \rightarrow \mathbb {C}$ is the function defined by

h(x) = \cos x+i \sin x

Let $\nu$ be the complex measure defined by $d \nu = h d \lambda$ .

(a)

Show that $\left\| \nu \right\| = t$ .

(b)

Show that if $E_{1}, E_{2}, \ldots$ is a sequence of disjoint Borel subsets of $[0, t]$ , then

\sum _{k = 1}^{\infty }\left|\nu \left(E_{k}\right)\right| < t

Remark.

This exercise shows that the supremum in the definition of $\left|\nu \right|([0, t])$ is not attained, even if countably many disjoint sets are allowed.

Problem 9A.9

Give an example to show that 9.9 can fail if the hypothesis that $\nu$ is a real measure is replaced by the hypothesis that $\nu$ is a complex measure.

Problem 9A.10

Suppose $(X, \mathcal{S})$ is a measurable space with $\mathcal{S} \neq \left\{ \varnothing , X\right\}$ . Prove that the total variation norm on $\mathcal{M}_{\mathrm{F}}(\mathcal{S})$ does not come from an inner product. In other words, show that there does not exist an inner product $\left\langle \cdot , \cdot \right\rangle$ on $\mathcal{M}_{\mathrm{F}}(\mathcal{S})$ such that $\left\| \nu \right\|_{\mathbf{F}} = \left\langle \nu , \nu \right\rangle^{1 / 2}$ for all $\nu \in \mathcal{M}(\mathcal{S})$ , where $\left\| \cdot \right\|_{\mathbf{F}}$ is the usual total variation norm on $\mathcal{M}(\mathcal{S})$ .

Problem 9A.11

(a)(b)

For $(X, \mathcal{S})$ a measurable space and $b \in X$ , define a finite (positive) measure $\delta_{b}$ on $(X, \mathcal{S})$ by

\delta _{b}(E) = \begin{cases} 1 & \text{ if } b \in E \\ 0 & \text{ if } b \notin E\end{cases}

for $E \in \mathcal{S}$ .

(a)

Show that if $b, c \in X$ , then $\left\| \delta_{b}+\delta_{c}\right\| = 2$ .

(b)

Give an example of a measurable space $(X, \mathcal{S})$ and $b, c \in X$ with $b \neq c$ such that $\left\| \delta_{b}-\delta_{c}\right\| \neq 2$ .

Section 9BDecomposition Theorems[18/83%]

Problem 9B.1

Suppose $\nu$ is a real measure on a measurable space $(X, \mathcal{S})$ . Prove that the Hahn decomposition of $\nu$ is almost unique, in the sense that if $A, B$ and $A^{\prime }, B^{\prime }$ are pairs satisfying the Hahn Decomposition Theorem (9.23), then

\left|\nu \right|\left(A \backslash A^{\prime }\right) = \left|\nu \right|\left(A^{\prime } \backslash A\right) = \left|\nu \right|\left(B \backslash B^{\prime }\right) = \left|\nu \right|\left(B^{\prime } \backslash B\right) = 0

Problem 9B.2

Suppose $\mu$ is a (positive) measure and $g, h \in \mathcal{L}^{1}(\mu )$ . Prove that $g \; d \mu \perp h \; d \mu$ if and only if $g(x) h(x) = 0$ for almost every $x \in X$ .

Problem 9B.3

(a)(b)(c)

Suppose $\nu$ and $\mu$ are complex measures on a measurable space $(X, \mathcal{S})$ . Show that the following are equivalent:

(a)

$\nu \perp \mu$ .

(b)

$\left|\nu \right| \perp \left|\mu \right|$ .

(c)

$\operatorname {Re} \nu \perp \mu$ and $\operatorname {Im} \nu \perp \mu$ .

Problem 9B.4

Suppose $\nu$ and $\mu$ are complex measures on a measurable space $(X, \mathcal{S})$ . Prove that if $\nu \perp \mu$ , then $\left|\nu +\mu \right| = \left|\nu \right|+\left|\mu \right|$ and $\left\| \nu +\mu \right\| = \left\| \nu \right\| +\left\| \mu \right\|$ .

Problem 9B.5

Suppose $\nu$ and $\mu$ are finite (positive) measures on a measurable space $(X, \mathcal{S})$ . Prove that $\nu \perp \mu$ if and only if $\left\| \nu -\mu \right\| = \left\| \nu \right\| +\left\| \mu \right\|$ .

Problem 9B.6

Suppose $\mu$ is a complex or positive measure on a measurable space $(X, \mathcal{S})$ . Prove that

\left\{ \nu \in \mathcal{M}_{\mathbf{F}}(\mathcal{S}): \nu \perp \mu \right\}

is a closed subspace of $\mathcal{M}_{\mathrm{F}}(\mathcal{S})$ .

Problem 9B.7

Use the Cantor set to prove that there exists a (positive) measure $\nu$ on $(\mathbb {R}, \mathcal{B})$ such that $\nu \perp \lambda$ and $\nu (\mathbb {R}) \neq 0$ but $\nu (\left\{ x\right\} ) = 0$ for every $x \in \mathbb {R}$ ; here $\lambda$ denotes Lebesgue measure on the $\sigma$ -algebra $\mathcal{B}$ of Borel subsets of $\mathbb {R}$ .

Remark.

Example 9.29 (second bullet point): let $r_{1}, r_{2}, \ldots$ be a list of the rational numbers, and suppose $w_{1}, w_{2}, \ldots$ is a bounded sequence of complex numbers. Define a complex measure $\nu$ on $(\mathbb {R}, \mathcal{B})$ by $\nu (E) = \sum_{\left\{ k \in \mathbb {N}: r_{k} \in E\right\} } \frac{w_{k}}{2^{k}}$ for $E \in \mathcal{B}$ . Then $\nu \perp \lambda$ because $\nu$ lives on $\mathbb {Q}$ and $\lambda$ lives on $\mathbb {R} \setminus \mathbb {Q}$ .

The second bullet point in Example 9.29 does not provide an example of the desired behavior because in that example, $\nu \left(\left\{ r_{k}\right\} \right) \neq 0$ for all $k \in \mathbb {N}$ with $w_{k} \neq 0$ .

Problem 9B.8

Suppose $\nu$ is a real measure on a measurable space $(X, \mathcal{S})$ . Prove that

\nu ^{+}(E) = \sup \left\{ \nu (D): D \in \mathcal{S} \text{ and } D \subseteq E\right\}

and

\nu ^{-}(E) = -\inf \left\{ \nu (D): D \in \mathcal{S} \text{ and } D \subseteq E\right\}

for all $E \in \mathcal{S}$ .

Problem 9B.9

Suppose $\mu$ is a (positive) finite measure on a measurable space $(X, \mathcal{S})$ and $h$ is a nonnegative function in $\mathcal{L}^{1}(\mu )$ . Thus $h \; d \mu \ll d \mu$ . Find a reasonable condition on $h$ that is equivalent to the condition $d \mu \ll h \; d \mu$ .

Problem 9B.10

(a)(b)(c)

Suppose $\mu$ is a (positive) measure on a measurable space $(X, \mathcal{S})$ and $\nu$ is a complex measure on $(X, \mathcal{S})$ . Show that the following are equivalent:

(a)

$\nu \ll \mu$ .

(b)

$\left|\nu \right| \ll \mu$ .

(c)

$\operatorname {Re} \nu \ll \mu$ and $\operatorname {Im} \nu \ll \mu$ .

Problem 9B.11

Suppose $\mu$ is a (positive) measure on a measurable space $(X, \mathcal{S})$ and $\nu$ is a real measure on $(X, \mathcal{S})$ . Show that $\nu \ll \mu$ if and only if $\nu^{+} \ll \mu$ and $\nu^{-} \ll \mu$ .

Problem 9B.12

Suppose $\mu$ is a (positive) measure on a measurable space $(X, \mathcal{S})$ . Prove that

\left\{ \nu \in \mathcal{M}_{\mathbf{F}}(\mathcal{S}): \nu \ll \mu \right\}

is a closed subspace of $\mathcal{M}_{\mathrm{F}}(\mathcal{S})$ .

Problem 9B.13

Give an example to show that the Radon-Nikodym Theorem (9.36) can fail if the $\sigma$ -finite hypothesis is eliminated.

Problem 9B.14

(a)(b)(c)

Suppose $\mu$ is a (positive) $\sigma$ -finite measure on a measurable space $(X, \mathcal{S})$ and $\nu$ is a complex measure on $(X, \mathcal{S})$ . Show that the following are equivalent:

(a)

$\nu \ll \mu$ .

(b)

For every $\varepsilon > 0$ , there exists $\delta > 0$ such that $\left|\nu (E)\right| < \varepsilon$ for every set $E \in \mathcal{S}$ with $\mu (E) < \delta$ .

(c)

For every $\varepsilon > 0$ , there exists $\delta > 0$ such that $\left|\nu \right|(E) < \varepsilon$ for every set $E \in \mathcal{S}$ with $\mu (E) < \delta$ .

Problem 9B.15

Prove 9.42 in the case where $p = 1$ .

Remark.

With the extra hypothesis that $\mu$ is a $\sigma$ -finite (positive) measure.

Theorem 9.42 (dual space of $L^p(\mu )$ is $L^{p'}(\mu )$ ): suppose $\mu$ is a (positive) measure and $1 \leq p < \infty$ (with the additional hypothesis that $\mu$ is a $\sigma$ -finite measure if $p = 1$ ). For $h \in L^{p'}(\mu )$ , define $\varphi_h: L^p(\mu ) \rightarrow \mathbf{F}$ by $\varphi_h(f) = \int fh \, d\mu$ . Then $h \mapsto \varphi_h$ is a one-to-one linear map from $L^{p'}(\mu )$ onto $\left(L^p(\mu )\right)'$ . Furthermore, $\left\| \varphi_h\right\| = \left\| h\right\|_{p'}$ for all $h \in L^{p'}(\mu )$ .

Problem 9B.16

Explain where the proof of 9.42 fails if $p = \infty$ .

Remark.

Problem 9B.17

Prove that if $\mu$ is a (positive) measure and $1 < p < \infty$ , then $L^{p}(\mu )$ is reflexive.

Remark.

See the definition before Exercise 19 in Section 7B for the meaning of reflexive.

Problem 9B.18

Prove that $L^{1}(\mathbb {R})$ is not reflexive.