Probability with Measure

1.10 Exercises on Chapter 1

1.1 Let $S=\{1,2,3,4\}$. Show that the set $\mc {A}=\big \{\emptyset , \{1,2,3\},\{4\},\{1,2\},\{1,2,4\},\{3\},S \big \}$ is not a $\sigma $-field on $S$.
1.2 Let $\Sigma _{1}$ and $\Sigma _{2}$ be $\sigma $-fields of subsets of a set $S$. Note that
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\begin{align*} \Sigma _{1} \cap \Sigma _{2} &= \{A \subseteq S \- A \in \Sigma _{1}\text { and }A\in \Sigma _{2}\}, \\ \Sigma _{1} \cup \Sigma _{2} &= \{A \subseteq S \- A \in \Sigma _{1}\text { or }A\in \Sigma _{2}\}. \end{align*}
- (a) Show that $\Sigma _{1} \cap \Sigma _{2}$ is a $\sigma $-field.
- (b) Why is $\Sigma _{1} \cup \Sigma _{2}$ not in general a $\sigma $-field? Give an example to demonstrate this.
1.3 Let $(S,\Sigma )$ be a measurable space and let $X\in \Sigma $. Show that $\Sigma _X=\{A\cap X\-A\in \Sigma \}$ is a $\sigma $-field on $X$.
1.4
- (a) Let $(S,\Sigma ,m)$ be a measure space. Show that for all $A,B \in \Sigma $,
  - (i) $m(A \cup B) + m(A \cap B) = m(A) + m(B),$
  - (ii) $m(A \cup B) \leq m(A) + m(B).$
- (b) Use (a)(ii) to prove that if $A_{1}, A_{2}, \ldots , A_{n} \in \Sigma $ then $m\left (\bigcup _{i=1}^{n}A_{i}\right ) \leq \sum _{i=1}^{n}m(A_{i}).$
1.5 Let $(S,\Sigma ,m)$ be a measure space.
- (a) Let $k > 0$. Show that $km$ is also a measure on $(S, \Sigma )$ where for all $A \in \Sigma $,
  
  \[ (km)(A) = km(A).\]
  
  Hence show that if $m$ is a finite measure and $m(S)>0$, then $\P (A) = \frac {m(A)}{m(S)}$ defines a probability measure for $A \in \Sigma $.
- (b) Let $B \in \Sigma $. Show that $m_{B}(A) = m(A \cap B)$ for $A \in \Sigma $ defines a measure on $(S, \Sigma )$.
- (c) Suppose that $m$ is a finite measure and $m(B) > 0$. Deduce that $\P _{B}$ is a probability measure where
  
  \[\P _{B}(A) = \frac {m_{B}(A)}{m(B)}.\]
  
  How does this relate to the notion of conditional probability?
1.6 Show that ${\cal B}(\R )$ contains all closed intervals $[a, b]$, where $-\infty < a < b < \infty $.
1.7
- (a) Let $m$ be a finite measure on the measurable space $(S, \Sigma )$.
  - (i) Let $A\in \Sigma $. Show that $m(S\sc A) = m(S) - m(A)$.
  - (ii) Let $(A_n)_{n\in \N }$ be a decreasing sequence of sets in $\Sigma $. Show that $m(A_{n})\to m(\cap _j A_j)$ as $n\to \infty $.
    
    This question proves part 2 of Lemma 1.7.1. We have already proved part 1 and you should use part 1 to prove part 2.
- (b) Take $S=\N $, let $\Sigma =\mc {P}(\N )$ and let $m=\#$ be counting measure. Give an example of a decreasing sequence of subsets $A_n\sw \Sigma $ for which $m(\cap _j A_j)\neq \lim _{n\to \infty } m(A_n)$.
1.8 Let $(S,\Sigma ,m)$ be a measure space and for each $n\in \N $ let $E_n$ have full measure. Show that $\bigcap _{n=1}^\infty E_n$ has full measure.
1.9 Show that if $S$ is a set containing $n$ elements, then the power set ${\cal P}(S)$ contains $2^{n}$ elements.

Hint: How many subsets are there of size $r$, for a fixed $1 \leq r \leq n$? The binomial theorem may also be of some use.

Challenge Questions

1.10 Let $S$ be a finite set and $\Sigma $ be a $\sigma $-field on $S$. Consider the set
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\begin{equation} \tag {$\star $}\label {eq:atoms} \Pi =\{ A \in \Sigma \- \text {if $B\in \Sigma $ and $B\sw A$ then either $B=A$ or $B=\emptyset $}\}. \end{equation}
- (a) Show that $\Pi $ is a finite set.
- (b) Using (a), let us enumerate the elements of $\Pi $ as $\Pi =\{\Pi _1, \Pi _2,\cdots , \Pi _k\}$, where each $\Pi _i$ is distinct from the others.
  - (i) Show that $\Pi _i \cap \Pi _j= \emptyset $ for $i\neq j$. Hint: Could $\Pi _i \cap \Pi _j$ be an element of $\Pi $?
  - (ii) Show that $\cup _{i=1}^k \Pi _i =S$. Hint: If $C=S\setminus \cup _{i=1}^k \Pi _i$ is non-empty, is $C\in \Pi $?
  - (iii) Let $A\in \Sigma $. Show that
    
    \[A=\bigcup _{i\in I}\Pi _i\]
    
    where $I=\{i=1,\ldots ,k\- A\cap \Pi _i\neq \emptyset \}$.
1.11 Prove that both of the following claims are false.
- (a) The Cantor set $C$ contains an open interval $(a,b)\sw C$, where $a<b$.
- (b) If a Borel set has non-zero Lebesgue measure then it contains an open interval.