Math Problems Vault

Problem 4.1

Show that in the proof of Theorem 4.1.1, $\mu \left(A_{3}\right) = 0$ where $A_{3} = \left\{ g \notin [0,1]\right\}$ and $g$ satisfies (1.9).

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Problem 4.2

Verify that $\mu_{1 s}$ , defined in (1.14) and $\mu_{2}$ are singular.

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Problem 4.3

Let $\nu$ be the counting measure on $[0,1]$ and $\int_{[0,1]} h d \nu < \infty$ for some nonnegative function $h$ . Show that $B = \left\{ x: h(x) > 0\right\}$ is countable.

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Problem 4.4

Prove Proposition 4.1.2.

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Problem 4.5

Find the Lebesgue decomposition of $\mu$ w.r.t. $\nu$ and the RadonNikodym derivative $\frac{d \mu_{a}}{d \nu }$ in the following cases where $\mu_{a}$ is the absolutely continuous component of $\mu$ w.r.t. $\nu$ .

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(a)

$\mu = \mathscr {N}\left(0, 1\right), \nu = \operatorname {Exp}\left(1\right)$

(b)

$\mu = \operatorname {Exp}\left(1\right), \nu = \mathscr {N}\left(0, 1\right)$

(c)

$\mu = \mu_{1}+\mu_{2}$ , where $\mu_{1} = \mathscr {N}\left(0, 1\right), \mu_{2} = \operatorname {Pois}\left(1\right)$ and $\nu = \operatorname {Cauchy}\left(0,1\right)$ .

(d)

$\mu = \mu_{1}+\mu_{2}, \nu = \operatorname {Geom}\left(p\right), 0 < p < 1$ , where $\mu_{1} = \mathscr {N}\left(0, 1\right)$ and $\mu_{2} =$ Pois1.

(e)

$\mu = \mu_{1}+\mu_{2}, \nu = \nu_{1}+\nu_{2}$ where $\mu_{1} = \mathscr {N}\left(0, 1\right), \mu_{2} = \operatorname {Pois}\left(1\right)$ , $\nu_{1} = \operatorname {Cauchy}\left(0,1\right)$ and $\nu_{2} = \operatorname {Geom}\left(p\right), 0 < p < 1$ .

(f)

$\mu = \operatorname {Bin}\left(10, 1 / 2\right), \nu =$ Pois1.

The measures referred to above are defined in Tables 4.6.1 and 4.6.2, given at the end of this section.

Problem 4.6

(a)(b)

Let $(\Omega , \mathcal{F}, \mu )$ be a measure space and $f \in L^{1}(\Omega , \mathcal{F}, \mu )$ . Let

\nu _{f}(A) \equiv \int _{A} f d \mu \quad \text{ for all } \quad A \in \mathcal{F}

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(a)

Show that $\nu_{f}$ is a finite signed measure.

(b)

Show that $\left\| \nu \right\| = \int_{\Omega }\left|f\right| d \mu$ and for $A \in \mathcal{F}, \nu_{f}^{+}(A) = \int_{A} f^{+} d \mu$ , $\nu_{f}^{-}(A) = \int_{A} f d \mu$ , and $\left|\nu_{f}\right|(A) = \int_{A}\left|f\right|(d \mu )$ .

Problem 4.7

(a)(b)(c)

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(a)

Let $\mu_{1}$ and $\mu_{2}$ be two finite measures such that both are dominated by a $\sigma$ -finite measure $\nu$ . Show that the total variation measure of the signed measure $\mu \equiv \mu_{1}-\mu_{2}$ is given by $\left|\mu \right|(A) = \int_{A}\left|h_{1}-h_{2}\right| d \nu$ where for $i = 1,2, h_{i} = \frac{d \mu_{i}}{d \nu }$ .

(b)

Conclude that if $\mu_{1}$ and $\mu_{2}$ are two measures on a countable set $\Omega \equiv \left\{ \omega_{i}\right\}_{i \geq 1}$ with $\mathcal{F} \equiv \mathcal{P}(\Omega )$ , then $\left|\mu \right|(A) = \sum_{i \in A} \mid \mu_{1}\left(\omega_{i}\right)- \mu_{2}\left(\omega_{i}\right) \mid$ .

(c)

Show that if $\mu_{n}$ is the Binomial $\left(n, p_{n}\right)$ measure and $\mu$ is the Poisson $(\lambda )$ measure, $0 < \lambda < \infty$ , then as $n \rightarrow \infty ,\left|\mu_{n}-\mu \right|(\cdot ) \rightarrow 0$ uniformly on $\mathcal{P}\left(\mathbb {Z}_{+}\right)$ iff $n p_{n} \rightarrow \lambda$ . (Hint: Show that for each $i \in \mathbb {Z}_{+} \equiv \left\{ 0,1,2, \ldots \right\} , \mu_{n}(\left\{ i\right\} ) \rightarrow \mu (\left\{ i\right\} )$ and use Scheffe's theorem.)

Problem 4.8

Let $\nu$ be a finite signed measure on a measurable space $(\Omega , \mathcal{F})$ and let $\left|\nu \right|$ be the total variation measure corresponding to $\nu$ . Show that for any $B \in \Omega^{+}, B \subset \mathcal{F}$ ,

|\nu |(B) = \nu (B)

where $\Omega^{+}$ is as defined in (2.13).

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Problem 4.9

Show that the Banach space $\mathbb {S}$ of finite signed measures on $(\mathbb {N}, \mathcal{P}(\mathbb {N}))$ is isomorphic to $\ell_{1}$ , the Banach space of absolutely convergent sequences $\left\{ x_{n}\right\}_{n \geq 1}$ in $\mathbb {R}$ .

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Problem 4.10

(a)(b)

Let $\mu_{1}$ and $\mu_{2}$ be two probability measures on $(\Omega , \mathcal{F})$ .

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(a)

Show that

\left\| \mu _{1}-\mu _{2}\right\| = 2 \sup \left\{ \left|\mu _{1}(A)-\mu _{2}(A)\right|: A \in \mathcal{F}\right\}

(Hint: For any $A \in \mathcal{F},\left\{ A, A^{c}\right\}$ is a partition of $\Omega$ and so $\left\| \mu_{1}- \mu_{2} \right\| \geq \left|\mu_{1}(A)-\mu_{2}(A)\right|+\left|\mu_{1}\left(A^{c}\right)-\mu_{2}\left(A^{c}\right)\right| = 2\left|\mu_{1}(A)-\mu_{2}(A)\right|$ , since $\mu_{1}$ and $\mu_{2}$ are probability measures. For the opposite inequality, use the Hahn decomposition of $\Omega$ w.r.t. $\mu_{1}-\mu_{2}$ and the fact $\left\| \mu_{1}-\mu_{2}\right\| = \left|\left(\mu_{1}-\mu_{2}\right)\left(\Omega^{+}\right)\right|+\left|\left(\mu_{1}-\mu_{2}\right)\left(\Omega^{-}\right)\right|$ . $)$

(b)

Show that $\left\| \mu_{1}-\mu_{2}\right\|$ is also equal to

\sup \left\{ \left|\int f d \mu _{1}-\int f d \mu _{2}\right|: f \in B(\Omega , \mathbb {R})\right\}

where $B(\Omega , \mathbb {R})$ is the collection of all $\mathcal{F}$ -measurable functions from $\Omega$ to $\mathbb {R}$ such that $\sup \left\{ \left|f(\omega )\right|: \omega \in \Omega \right\} \leq 1$ .

Problem 4.11

(a)(b)(c)

Let $(\Omega , \mathcal{F})$ be a measurable space.

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(a)

Let $\left\{ \mu_{n}\right\}_{n \geq 1}$ be a sequence of finite measures on $(\Omega , \mathcal{F})$ . Show that there exists a probability measure $\lambda$ such that $\mu_{n} \ll \lambda$ . (Hint: Consider $\lambda (\cdot ) = \sum_{n = 1}^{\infty } \frac{1}{2^{n}} \frac{\mu_{n}(\cdot )}{\mu_{n}(\Omega )}.$ )

(b)

Extend (a) to the case where $\left\{ \mu_{n}\right\}_{n \geq 1}$ are $\sigma$ -finite.

(c)

Conclude that for any sequence $\left\{ \nu_{n}\right\}_{n \geq 1}$ of finite signed measures on $(\Omega , \mathcal{F})$ , there exists a probability measure $\lambda$ such that $\left|\nu_{n}\right| \ll \lambda$ for all $n \geq 1$ .

Problem 4.12

Let $\left\{ \mu_{n}\right\}_{n \geq 1}$ be a sequence of finite measures on a measurable space $(\Omega , \mathcal{F})$ . Show that there exists a finite measure $\mu$ on $(\Omega , \mathcal{F})$ such that $\left\| \mu_{n}-\mu \right\| \rightarrow 0$ iff there is a finite measure $\lambda$ dominating $\mu$ and $\mu_{n}$ , $n \geq 1$ such that the Radon-Nikodym derivatives $f_{n} \equiv \frac{d \mu_{n}}{d \lambda } \rightarrow f \equiv \frac{d \mu }{d \lambda }$ in measure on $(\Omega , \mathcal{F}, \lambda )$ and $\mu_{n}(\Omega ) \rightarrow \mu (\Omega )$ .

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Problem 4.13

(a)(b)(c)

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(a)

Let $\mu_{1}$ and $\mu_{2}$ be two finite measures on $(\Omega , \mathcal{F})$ . Let $\mu_{1} = \mu_{1 a}+ \mu_{1 s}$ be the Lebesgue-Radon-Nikodym decomposition of $\mu_{1}$ w.r.t. $\mu_{2}$ as in Theorem 4.1.1. Show that if $\mu = \mu_{1}-\mu_{2}$ , then for all $A \in \mathcal{F}$ ,

|\mu |(A) = \int _{A}|h-1| d \mu _{2}+\mu _{1 s}(A) \quad \text{ where } \quad h = \frac{d \mu _{1 a}}{d \mu _{2}}

is the Radon-Nikodym derivative of $\mu_{1 a}$ w.r.t. $\mu_{2}$ . Conclude that if $\mu_{1} \perp \mu_{2}$ , then $\left|\mu \right|(\cdot ) = \mu_{1}(\cdot )+\mu_{2}(\cdot )$ and if $\mu_{1} \ll \mu_{2}$ , then $\left|\mu \right|(A) = \int_{A}\left|\frac{d \mu_{1}}{d \mu_{2}}-1\right| d \mu_{2}$ .

(b)

Compute $\left|\mu \right|(\cdot ),\left\| \mu \right\|$ if $\mu = \mu_{1}-\mu_{2}$ for the following cases

$\mu_{1} = \mathscr {N}\left(0, 1\right), \mu_{2} = \mathscr {N}\left(1, 1\right)$
$\mu_{1} = \operatorname {Cauchy}\left(0,1\right), \mu_{2} = \mathscr {N}\left(0, 1\right)$
$\mu_{1} = \mathscr {N}\left(0, 1\right), \mu_{2} = \operatorname {Pois}\left(\lambda \right)$ .

(c)

Establish Proposition 4.2.7.

Problem 4.14

(a)(b)

Give another proof of the completeness of $(\mathbb {S},\left\| \cdot \right\| )$ by verifying the following steps.

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(a)

For any sequence $\left\{ \nu_{n}\right\}_{n \geq 1}$ in $\mathbb {S}$ , there is a finite measure $\lambda$ and $\left\{ f_{n}\right\}_{n \geq 1} \subset L^{1}(\Omega , \mathcal{F}, \lambda )$ such that

\nu _{n}(A) = \int _{A} f_{n} d \lambda \quad \text{ for all } \quad A \in \mathcal{F}, \quad \text{ for all } \quad n \geq 1

(b)

$\left\{ \nu_{n}\right\}_{n \geq 1}$ Cauchy in $\mathbb {S}$ is the same as $\left\{ f_{n}\right\}_{n \geq 1}$ Cauchy in $L^{1}(\Omega , \mathcal{F}, \lambda )$ and hence, the completeness of $(\mathbb {S},\left\| \cdot \right\| )$ follows from the completeness of $L^{1}(\Omega , \mathcal{F}, \lambda )$ .

Problem 4.15

(a)(b)(c)

Let $f, g \in B V[a, b]$ .

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(a)

Show that $P(f+g ;[a, b]) \leq P(f ;[a, b])+P(g ;[a, b])$ and that the same is true for $N(\cdot ; \cdot )$ and $T(\cdot ; \cdot )$ .

(b)

Show that for any $c \in \mathbb {R}$ ,

P(c f ;[a, b]) = \left|c\right| P(f ;[a, b])

and do the same for $N(\cdot ; \cdot )$ and $T(\cdot ; \cdot )$ .

(c)

For any $a < c < b, P(f ;[a, b]) = P(f ;[a, c])+P(f ;[c, b])$ .

Problem 4.16

Let $\left\{ f_{n}\right\}_{n \geq 1} \subset B V[a, b]$ and let $\lim_{n} f_{n}(x) = f(x)$ for all $x$ in $[a, b]$ . Show that $P(f ;[a, b]) \leq \lim_{n \rightarrow \infty } P(f_{n} ;[a, b])$ and do the same for $N(\cdot ; \cdot )$ and $T(\cdot ; \cdot )$ .

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Problem 4.17

Let $f \in B V[a, b]$ . Show that $f$ is continuous except on an at most countable set.

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Problem 4.18

Let $F:[a, b] \rightarrow \mathbb {R}$ be a.c. Show that it is of bounded variation.

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Problem 4.19

Let $F$ be an absolutely continuous nondecreasing function on $\mathbb {R}$ . Let $\mu_{F}$ be the Lebesgue-Stieltjes measure corresponding to $F$ . Show that for any $h \in L^{1}\left(\mathbb {R}, \mathcal{M}_{\mu_{F}}, \mu_{F}\right)$ ,

\int _{\mathbb {R}} h d \mu _{F} = \int h f d m

where $f$ is a nonnegative measurable function such that $F(b)-F(a) = \int_{[a, b]} f d m$ for any $a < b$ .

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Problem 4.20

Let $F:[a, b] \rightarrow \mathbb {R}$ be absolutely continuous with $F^{\prime }(\cdot ) > 0$ a.e. on $[a, b]$ , where $-\infty < a < b < \infty$ . Let $F(a) = c$ and $F(b) = d$ . Let $m(\cdot )$ denote the Lebesgue measure on $\mathbb {R}$ . Show the following:

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(a)

(Change of variables formula). For any $g:[c, d] \rightarrow \mathbb {R}$ and Lebesgue measurable and integrable w.r.t. $m$

\int _{[c, d]} g d m = \int _{[a, b]} g(F) F^{\prime } d m

(b)

For any Borel set $E \subset [a, b], F(E)$ is also a Borel set.

(c)

$\nu (\cdot ) \equiv m(F(\cdot ))$ is a measure on $\mathcal{B}([a, b])$ and $\nu \ll m$ with

\frac{d \nu }{d m}(\cdot ) = F^{\prime }(\cdot )

(d)

$\lambda (\cdot ) \equiv m F^{-1}(\cdot )$ is a measure on $\mathcal{B}([c, d])$ and $\lambda \ll m$ with

\frac{d \lambda }{d m}(\cdot ) = \left(F^{\prime }\left(F^{-1}(\cdot )\right)\right)^{-1}

(e)

For any measure $\mu \ll m$ on $\mathcal{B}([a, b])$ the measure $\psi (\cdot ) \equiv \mu F^{-1}(\cdot )$ is dominated by $m$ with

\frac{d \psi }{d m} = \frac{d \mu }{d m}\left(F^{-1}(\cdot )\right)\left(F^{\prime }\left(F^{-1}(\cdot )\right)\right)^{-1}

(f)

Establish (a) assuming that $g$ and $F^{\prime }$ are both continuous noting that both integrals reduce to Riemann integrals.

Problem 4.21

(a)(b)(c)

Let $F: \mathbb {R} \rightarrow \mathbb {R}$ be absolutely continuous on every finite interval.

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(a)

Show that the $f$ in (4.1) can be chosen independently of the interval $[a, b]$ .

(b)

Further, if $f$ is integrable over $\mathbb {R}$ , then $\lim_{x \rightarrow -\infty } F(x) \equiv F(-\infty )$ and $\lim_{x \rightarrow \infty } F(x) \equiv F(\infty )$ exist and $F(x) = F(-\infty )+ \int_{(-\infty , x)} f d \mu_{L}$ for all $x$ in $\mathbb {R}$ .

(c)

Give an example where $F: \mathbb {R} \rightarrow \mathbb {R}$ is a.c., but $f$ is not integrable over $\mathbb {R}$ .

Problem 4.22

(a)(b)(c)

Let $F: \mathbb {R} \rightarrow \mathbb {R}$ be absolutely continuous on bounded intervals. Let $\left\{ I_{j} = 1 \leq j \leq k \leq \infty \right\}$ be a collection of disjoint intervals such that $\bigcup_{j = 1}^{k} I_{j} \equiv \mathbb {R}$ and on each $I_{j}, F^{\prime }(\cdot ) > 0$ a.e. or $F^{\prime }(\cdot ) < 0$ a.e. w.r.t. $m$ .

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(a)

Show that for any $h \in L^{1}(\mathbb {R}, m)$ ,

\int _{\mathbb {R}} h d m = \int _{\mathbb {R}} h(F(\cdot ))\left|F^{\prime }(\cdot )\right| d m.

(b)

Show that if $\mu$ is a measure on $(\mathbb {R}, \mathcal{B}(\mathbb {R}))$ dominated by $m$ then the measure $\mu F^{-1}(\cdot )$ is also dominated by $m$ and

\frac{d \mu F^{-1}}{d m}(y) = \sum _{x_{j} \in D(y)} \frac{f\left(x_{j}\right)}{\left|F^{\prime }\left(x_{j}\right)\right|}

where $f(\cdot ) = \frac{d \mu }{d m}$ and $D(y) = \left\{ x_{j}: x_{j} \in I_{j}, F\left(x_{j}\right) = y\right\}$ .

(c)

Let $\mu$ be the $\mathscr {N}\left(0, 1\right)$ measure on $(\mathbb {R}, \mathcal{B}(\mathbb {R}))$ , i.e.,

\frac{d \mu }{d m}(x) = \frac{1}{\sqrt{2 n}} e^{-\frac{x^{2}}{2}},-\infty < x < \infty

Let $F(x) = x^{2}$ . Find $\frac{d \mu F^{-1}}{d m}$ .

Problem 4.23

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be integrable w.r.t. $m$ on bounded intervals. Show that for almost all $x_{0}$ in $\mathbb {R}$ (w.r.t. $m$ ),

\lim _{\substack {a \uparrow x_{0} \\ b \downarrow x_{0}}} \frac{1}{(b-a)} \int _{a}^{b}\left|f(x)-f\left(x_{0}\right)\right| d x = 0.

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Problem 4.24

Use the hint to the above problem to establish Theorem 4.4.5.

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Problem 4.25

Let $(\Omega , \mathcal{B})$ be a measurable space. Let $\left\{ \mu_{n}\right\}_{n \geq 1}$ and $\mu$ be $\sigma$ -finite measures on $(\Omega , \mathcal{B})$ . Let for each $n \geq 1, \mu_{n} = \mu_{n a}+\mu_{n s}$ be the Lebesgue decomposition of $\mu_{n}$ w.r.t. $\mu$ with $\mu_{n a} \ll \mu$ and $\mu_{n s} \perp \mu$ . Let $\lambda = \sum_{n \geq 1} \mu_{n}, \lambda_{a} = \sum_{n \geq 1} \mu_{n a}, \lambda_{s} = \sum_{n \geq 1} \mu_{n s}$ . Show that $\lambda_{a} \ll \mu$ and $\lambda_{s} \perp \mu$ and that $\lambda = \lambda_{a}+\lambda_{s}$ is the Lebesgue decomposition of $\lambda$ w.r.t. $\mu$ .

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Problem 4.26

Let $\left\{ \mu_{n}\right\}_{n \geq 1}$ be Radon measures on $\left(\mathbb {R}^{k}, \mathcal{B}\left(\mathbb {R}^{k}\right)\right)$ and $m$ be the Lebesgue measure on $\mathbb {R}^{k}$ . Show that if $\lambda = \sum_{n = 1}^{\infty } \mu_{n}$ is also a Radon measure, then

D \lambda = \sum _{n = 1}^{\infty } D \mu _{n} \quad \text{ a.e. } \quad (m)

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Problem 4.27

Let $F_{n}, n \geq 1$ be a sequence of nondecreasing functions from $\mathbb {R} \rightarrow \mathbb {R}$ . Let $F(x) = \sum_{n \geq 1} F_{n}(x) < \infty$ for all $x \in \mathbb {R}$ . Show that $F(\cdot )$ is nondecreasing and

F^{\prime }(\cdot ) = \sum _{n \geq 1} F_{n}^{\prime }(\cdot ) \quad \text{ a.e. } \quad (m)

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Problem 4.28

Let $E$ be a Lebesgue measurable set in $\mathbb {R}$ . The metric density of $E$ at $x$ is defined as

D_{E}(x) \equiv \lim _{\delta \downarrow 0} \frac{m(E \cap (x-\delta , x+\delta ))}{2 \delta }

if it exists. Show that $D_{E}(\cdot ) = I_{E}(\cdot )$ a.e. $m$ .

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Problem 4.29

Let $F, G:[a, b] \rightarrow \mathbb {R}$ be both absolutely continuous. Show that $H = F G$ is also absolutely continuous on $[a, b]$ and that

\int _{[a, b]} F d G+\int _{[a, b]} G d F = F(b) G(b)-F(a) G(a).

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Problem 4.30

(a)(b)(c)(d)

Let $(\Omega , \mathcal{F}, \mu )$ be a finite measure space. Fix $1 \leq p < \infty$ . Let $T$ : $L^{p}(\mu ) \rightarrow \mathbb {R}$ be a bounded linear functional as defined in (3.2.10) (cf. Section 3.2). Complete the following outline of a proof of Theorem 3.2.3 (Riesz representation theorem).

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(a)

Let $\nu (A) \equiv T\left(I_{A}\right), A \in \mathcal{F}$ . Verify that $\nu (\cdot )$ is a signed measure on $(\Omega , \mathcal{F})$ .

(b)

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

(c)

Let $g \equiv \frac{d \nu }{d \mu }$ . Show that $g \in L^{q}(\mu )$ where $q = \frac{p}{p-1}, 1 < p < \infty$ and $q = \infty$ if $p = 1$ .

(d)

Show that $T = T_{g}$ .

Problem 4.31

Prove Theorem 4.5.2 and Corollary 4.5.3.

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Problem 4.32

For $0 < \alpha < 1$ , construct a Cantor like set $C_{\alpha }$ as described in Remark 4.5.1.

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Problem 4.33

Show that the Cantor ternary function can be expressed as in (5.5).

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Problem 4.34

(a)(b)

Let $(\Omega , \mathcal{F}, \mu )$ be a $\sigma$ -finite measure space.

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(a)

Let $\mathcal{G}$ be a $\sigma$ -algebra $\subset \mathcal{F}$ . Let $f: \Omega \rightarrow \mathbb {R}_{+}$ be $\langle \mathcal{F}, \mathcal{B}(\mathbb {R})\rangle$ measurable. Show that there exists a $g: \Omega \rightarrow \mathbb {R}_{+}$ that is $\langle \mathcal{G}, \mathcal{B}(\mathbb {R})\rangle$ -measurable and $\nu (A) \equiv \int_{A} f d \mu = \int_{A} g d \mu$ for all $A$ in $\mathcal{G}$ . (Hint: Apply Theorem 4.1.1 (b) to the measures $\nu$ and $\mu$ restricted to $\mathcal{G}$ . When $\mu$ is a probability measure, $g$ is called the conditional expectation of $f$ given $\mathcal{G}$ (cf. Chapter 12).)

(b)

Now suppose $\mathcal{G} = \sigma \left\langle \left\{ A_{i}\right\}_{i \geq 1}\right\rangle$ where $\left\{ A_{i}\right\}_{i \geq 1}$ is a partition of $\Omega \subset \mathcal{F}$ . Determine $g(\cdot )$ explicitly on each $A_{i}$ such that $0 < \mu \left(A_{i}\right) < \infty$ .