Measure, Integral and Probability

Marek Capi?ski, Ekkehard Kopp

Chapter 7 The Radon-Nikodym Theorem - all with Video Answers

Educators

Chapter Questions

Problem 1

Let $\lambda_{1}, \lambda_{2}$ and $\mu$ be measures on $(\Omega, \mathcal{F})$. Show that if $\lambda_{1} \ll \mu$ and $\lambda_{2} \ll \mu$ then $\left(\lambda_{1}+\lambda_{2}\right) \ll \mu$.

Victor Salazar

Numerade Educator

03:04

Let $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ be finite partitions of $\Omega$. Show that the coarsest partition (i.e. with least number of sets) which refines them both consists of all intersections $A \cap B$, where $A \in \mathcal{P}_{1}, B \in \mathcal{P}_{2}$.

Doruk Isik

Numerade Educator

03:29

Problem 3

Let $\Omega=[0,1]$ with Lebesgue measure and consider measures $\mu, \nu$ given by densities $\mathbf{1}_{A}, \mathbf{1}_{B}$ respectively. Find a condition on the sets $A, B$ so that $\mu$ dominates $\nu$ and find the Radon-Nikodym derivative $\frac{d \nu}{d \mu}$ applying the above definition of the function $h$.

Nick Johnson

Numerade Educator

04:02

Problem 4

Suppose $\Omega$ is a finite set equipped with the algebra of all subsets. Let $\mu$ and $\nu$ be two measures on $\Omega$ such that $\mu(\{\omega\}) \neq 0, \nu(\{\omega\}) \neq 0$, for all $\omega \in \Omega$. Decide under which conditions $\mu$ dominates $\nu$ and find $\frac{d \nu}{d \mu}$.

Mengchun Cai

Numerade Educator

01:03

Problem 5

Let $\Omega=[0,1]$ with Lebesgue measure and consider probability measures $\mu, \nu$ given by densities $f, g$ respectively. Find a condition characterising the absolute continuity $\nu \ll \mu$ and find the Radon-Nikodym derivative $\frac{d \nu}{d \mu}$

Ahmed Ibrahim

Numerade Educator

04:02

Problem 6

Suppose $\Omega$ is a finite set equipped with the algebra of all subsets and let $\mu$ and $\nu$ be two measures on $\Omega$. Characterise the absolute continuity $\nu \ll \mu$ and find $\frac{d \nu}{d \mu}$.

Mengchun Cai

Numerade Educator

01:49

Problem 7

Show that if $\mu, \nu$ are equivalent measures, i.e. both $\nu \ll \mu$ and $\mu \ll \nu$ are true, then
$$
\frac{d \mu}{d \nu}=\left(\frac{d \nu}{d \mu}\right)^{-1} \text { a.s. }(\mu).
$$

Nick Johnson

Numerade Educator

04:00

Problem 8

Consider the following measures on the real line: $P_{1}=\delta_{0}, P_{2}=$ $\left.\frac{1}{25} m\right|_{[0,25]}, P_{3}=\frac{1}{2} P_{1}+\frac{1}{2} P_{2}$ (see Example 3.1). For which $i \neq j$ do we have $P_{i} \ll P_{j}$ ? Find the Radon-Nikodym derivative in each such case.

Yujie Wang

College of San Mateo

01:48

Problem 9

Let $\lambda=\delta_{0}+\left.m\right|_{[1,3]}, \mu=\delta_{1}+\left.m\right|_{[2,4]}$ and find $\lambda_{a}, \lambda_{s}$, and $h$ as in Corollary 7.10.

Varsha Aggarwal

Numerade Educator

01:33

Problem 10

Suppose the monotone increasing function $F$ is non-constant at most countably many points (as would be the case for a discrete distribution). Show that every subset of $\mathbb{R}$ is $m_{F}$-measurable.

Tanishq Gupta

Numerade Educator

05:24

Problem 11

Find the Lebesgue-Stieltjes measure $m_{F}$ generated by
$$
F(x)=\left\{\begin{array}{l}
0 \text { if } x<0 \\
2 x \text { if } x \in[0,1] \\
2 \text { if } x \geq 1
\end{array}\right.
$$

Joseph David

Numerade Educator

03:17

Problem 12

Decide which of the following functions are absolutely continuous: (a) $f(x)=|x|, x \in[-1,1]$, (b) $g(x)=\sqrt{x}, x \in[0,1],(\mathrm{c})$ the Lebesgue function.

Faizanullah Kazmi

Numerade Educator

07:42

Problem 13

(a) Let $F$ be monotone increasing on $[a, b]$. Find $T_{F}[a, b]$.
(b) Prove that if $F \in B V[a, b]$ then $F$ is continuous a.e. $(m)$ and Lebesgue-measurable.
(c) Find a differentiable function which is not in $B V[0,1]$.
(d) Show that if there is a (Lipschitz) constant $M>0$ such that $\mid F(x)-$ $F(y)|\leq M| x-y \mid$ for all $x, y \in[a, b]$, then $F \in B V[a, b] .$

Regina Hays

Numerade Educator

02:03

Problem 14

Let $\nu$ be a bounded signed measure. Show that for all $F, \nu^{+}(F)=$ $\sup _{G \subset F} \nu(G), \nu^{-}(F)=-\inf _{G \subset F} \nu(G)$, all the sets concerned being members of $\mathcal{F}$.

James Chok

Numerade Educator

04:28

Problem 15

Show that when $\nu(F)=\int_{F} f \mathrm{~d} \mu$ where $f \in \mathcal{L}^{1}(\mu)$, where $\mu$ is a (positive) measure, the Hahn decomposition sets are $A=\{f<0\}$ and $B=\{f \geq$ $0\}$, and $\nu^{+}(F)=\int_{F} f^{+} \mathrm{d} \nu$, while $\nu^{-}(F)=\int_{F} f^{-} \mathrm{d} \nu$

Foster Wisusik

Numerade Educator

03:29

Problem 16

Verify the following: Let $\mu$ be a finite measure and define the signed measure $\nu$ by $\nu(F)=\int_{F} g \mathrm{~d} \mu$. Prove that $f \in L^{1}(\nu)$ if and only if $f g \in L^{1}(\mu)$ and $\int_{E} f \mathrm{~d} \nu=\int_{E} f g \mathrm{~d} \mu$ for all $\mu$-measurable sets $E$

Nick Johnson

Numerade Educator

07:47

Problem 17

Let $\Omega=[0,1]$ with Lebesgue measure and let $X(\omega)=\omega$. Find $\mathbb{E}(X \mid \mathcal{G})$ if (a) $\mathcal{G}=\left\{\left[0, \frac{1}{2}\right],\left(\frac{1}{2}, 1\right],[0,1], \varnothing\right\}$, (b) $\mathcal{G}$ is generated by the family of sets $\left\{B \subset\left[0, \frac{1}{2}\right]\right.$, Borel $\}$

Mengchun Cai

Numerade Educator

04:48

Problem 18

Let $Z_{n} \geq 0$ be a sequence of independent random variables with $\mathbb{E}\left(Z_{n}\right)= \mu=1$ Let $\mathcal{F}_{n}=\sigma\left\{Z_{k}: k \leq n\right\}$ and show that, $X_{0}=1, X_{n}=Z_{1} Z_{2} \ldots Z_{n}$ $(n \geq 1)$ defines a martingale for $\left(\mathcal{F}_{n}\right)$, provided all the products are integrable random variables, which holds, e.g., if all $Z_{n} \in \mathcal{L}^{\infty}(\Omega, \mathcal{F}, P)$.

Abhirup Pal

Numerade Educator

02:36

Problem 19

Let $\left(Z_{n}\right)_{n \geq 1}$ be a sequence of independent random variables with mean $\mu=\mathbb{E}\left(Z_{n}\right) \neq 0$ for all $n$. Show that the sequence of their partial sums $X_{n}=Z_{1}+Z_{2}+\cdots+Z_{n}$ is not a martingale for the filtration $\left(\mathcal{F}_{n}\right)_{n}$ where $\mathcal{F}_{n}=\sigma\left\{Z_{k}: k \leq n\right\}$. How can we 'compensate' for this by altering $X_{n} ?$

Amany Waheeb

Numerade Educator

01:36

Problem 20

Suppose $\left(Z_{n}\right)_{n \geq 1}$ is a sequence of Bernoulli random variables, with each $Z_{n}$ taking the values 1 and $-1$, each with probability $\frac{1}{2} .$ Let $X_{0}=0$, $X_{n}=Z_{1}+Z_{2}+\cdots+Z_{n}$, and let $\left(\mathcal{F}_{n}\right)_{n}$ be the natural filtration generated by the $\left(Z_{n}\right)$. Verify that $\left(X_{n}^{2}\right)$ is a submartingale, and find the increasing process $\left(A_{n}\right)$ in its Doob decomposition. What 'unexpected' property of $\left(A_{n}\right)$ can you detect in this example?

Adriano Chikande

Numerade Educator

01:17

Problem 21

Show that $\exp \{-r t\} S(t)$ is a martingale.

Bryan Lynn

Numerade Educator