Probability Measures

Suppose that $\Omega \subseteq \mathscr {F}$ and that $A, B \in \mathscr {F}$ implies $A-B=A \cap B^{c} \in \mathscr {F}$. Show that $\mathscr {F}$ is a field.

Suppose that $\Omega \in \mathscr {F}$ and that $\mathscr {F}$ is closed under the formation of complements and finite disjoint unions. Show that $\mathscr {F}$ need not be a field.

Suppose that $\mathscr {F}_{n}$ are fields satisfying $\mathscr {F}_{n} \subset \mathscr {F}_{n+1}$. Show that $\bigcup_{n=1}^{\infty } \mathscr {F}_{n}$ is a field.

Suppose that $\mathscr {F}_{n}$ are $\sigma$-fields satisfying $\mathscr {F}_{n} \subset \mathscr {F}_{n+1}$. Show by example that $\bigcup_{n=1}^{\infty } \mathscr {F}_{n}$ need not be a $\sigma$-field.

Show that $f(\mathscr {A})$ is indeed a field, that $\mathscr {A} \subset f(\mathscr {A})$, and that $f(\mathscr {A})$ is minimal in the sense that if $\mathscr {G}$ is a field and $\mathscr {A} \subset \mathscr {G}$, then $f(\mathscr {A}) \subset \mathscr {G}$.

Show that for nonempty $\mathscr {A}, f(\mathscr {A})$ is the class of sets of the form $\bigcup_{i=1}^{m} \bigcap_{j=1}^{n_{i}} A_{i j}$, where for each $i$ and $j$ either $A_{i j} \in \mathscr {A}$ or $A_{i j}^{c} \in \mathscr {A}$, and where the $m$ sets $\bigcap_{j=1}^{n_{i}} A_{i j}, 1 \leq i \leq m$, are disjoint. The sets in $f(\mathscr {A})$ can thus be explicitly presented, which is not in general true of the sets in $\sigma (\mathscr {A})$.

Show that if $\mathscr {A}$ consists of the singletons, then $f(\mathscr {A})$ is the field in Example 2.3.

Show that $f(\mathscr {A}) \subset \sigma (\mathscr {A})$, that $f(\mathscr {A})=\sigma (\mathscr {A})$ if $\mathscr {A}$ is finite, and that $\sigma (f(\mathscr {A}))=\sigma (\mathscr {A})$.

Show that if $\mathscr {A}$ is countable, then $f(\mathscr {A})$ is countable.

Show for fields $\mathscr {F}_{1}$ and $\mathscr {F}_{2}$ that $f\left(\mathscr {F}_{1} \cup \mathscr {F}_{2}\right)$ consists of the finite disjoint unions of sets $A_{1} \cap A_{2}$ with $A_{i} \in \mathscr {F}_{i}$. Extend.

Show that if $\sigma (\mathscr {A})$ contains every subset of $\Omega$, then for each pair $\omega$ and $\omega^{\prime }$ of distinct points in $\Omega$ there is in $\mathscr {A}$ an $A$ such that $I_{A}(\omega ) \neq I_{A}\left(\omega^{\prime }\right)$

Show that the reverse implication holds if $\Omega$ is countable.

Show by example that the reverse implication need not hold for uncountable $\Omega$

Show that the $\sigma$-field $\mathscr {B}$ of Borel sets is countably generated.

Show that the $\sigma$-field of Example 2.4 is countably generated if and only if $\Omega$ is countable.

Suppose that $\mathscr {F}_{1}$ and $\mathscr {F}_{2}$ are $\sigma$-fields, $\mathscr {F}_{1} \subset \mathscr {F}_{2}$, and $\mathscr {F}_{2}$ is countably generated. Show by example that $\mathscr {F}_{1}$ may not be countably generated.

Let $\mathscr {F}$ be the field consisting of the finite and the cofinite sets in an infinite $\Omega$, and define $P$ on $\mathscr {F}$ by taking $\mathbb {P}\left(A\right)$ to be 0 or 1 as $A$ is finite or cofinite. (Note that $P$ is not well defined if $\Omega$ is finite.) Show that $P$ is finitely additive.

Show that this $P$ is not countably additive if $\Omega$ is countably infinite.

Show that this $P$ is countably additive if $\Omega$ is uncountable.

Now let $\mathscr {F}$ be the $\sigma$-field consisting of the countable and the cocountable sets in an uncountable $\Omega$, and define $P$ on $\mathscr {F}$ by taking $\mathbb {P}\left(A\right)$ to be 0 or 1 as $A$ is countable or cocountable. (Note that $P$ is not well defined if $\Omega$ is countable.) Show that $P$ is countably additive.

Suppose that $P$ is a probability measure on a field $\mathscr {F}$. Suppose that $A_{t} \in \mathscr {F}$ for $t>0$, that $A_{s} \subset A_{t}$ for $s<t$, and that $A=\cup_{t>0} A_{t} \in \mathscr {F}$. Extend Theorem 2.1(i) by showing that $\mathbb {P}\left(A_{t}\right) \uparrow \mathbb {P}\left(A\right)$ as $t \rightarrow \infty$. Show that $A$ necessarily lies in $\mathscr {F}$ if it is a $\sigma$-field.

Extend Theorem 2.1(ii) in the same way.

Show that $D$ is finitely but not countably additive on $\mathscr {D}$.

Show that $\mathscr {D}$ contains the empty set and $\Omega$ and is closed under the formation of complements, proper differences, and finite disjoint unions, but is not closed under the formation of countable disjoint unions or of finite unions that are not disjoint.

Let $\mathscr {M}$ consist of the periodic sets $M_{a}=[k a: k=1,2, \ldots ]$. Observe that

Show that the field $f(\mathscr {M})$ generated by $\mathscr {M}$ (see Problem 2.5) is contained in $\mathscr {D}$. Show that $D$ is completely determined on $f(\mathscr {M})$ by the value it gives for each $a$ to the event that $m$ is divisible by $a$.

Assume that $\sum p^{-1}$ diverges (sum over all primes; see Problem 5.20(e)) and prove that $D$, although finitely additive, is not countably additive on the field $f(M)$.

Euler's function $\varphi (n)$ is the number of positive integers less than $n$ and relatively prime to it. Let $p_{1}, \ldots , p_{r}$ be the distinct prime factors of $n$; from the inclusion-exclusion formula for the events [ $m: p_{i} \mid m$ ], (2.36), and the fact that the $p_{i}$ divide $n$, deduce

Show for $0 \leq x \leq 1$ that $D(A)=x$ for some $A$.

Show that $D$ is translation invariant: If $B=[m+1: m \in A]$, then $B$ has a density if and only if $A$ does, in which case $D(A)=D(B)$.

Assuming the existence of Lebesgue measure $\lambda$ on $\mathscr {B}$, prove that it is nonatomic.

Show in the nonatomic case that $\mathbb {P}\left(A\right)>0$ and $\epsilon >0$ imply that there exists a $B$ such that $B \subset A$ and $0<\mathbb {P}\left(B\right)<\epsilon$.

Show in the nonatomic case that $0 \leq x \leq \mathbb {P}\left(A\right)$ implies that there exists a $B$ such that $B \subset A$ and $\mathbb {P}\left(B\right)=x$. Hint: Inductively define classes $\mathscr {Z}_{n}$, numbers $h_{n}$, and sets $H_{n}$ by $\mathscr {H}_{0}=\left\{ \varnothing \right\} =\left\{ H_{0}\right\} , \quad \mathscr {H}_{n}=\left[H: H \subset A-\bigcup_{k<n} H_{k}\right.$, $\left.\mathbb {P}\left(\cup_{k<n} H_{k}\right)+\mathbb {P}\left(H\right) \leq x\right], h_{n}=\sup \left[\mathbb {P}\left(H\right) \cdot H \in \mathscr {U}_{n}\right]$, and $\mathbb {P}\left(H_{n}\right)>h_{n}-n^{-1}$. Consider $\bigcup_{k} H_{k}$.

Show in the nonatomic case that, if $p_{1}, p_{2}, \ldots$ are nonnegative and add to 1, then $A$ can be decomposed into sets $B_{1}, B_{2}, \ldots$ such that $\mathbb {P}\left(B_{n}\right)=p_{n} \mathbb {P}\left(A\right)$.

Suppose that $\mathscr {A}=\left\{ A_{1}, A_{2}, \ldots \right\}$ is a countable partition of $\Omega$ Show (see (2.27)) that $\mathscr {A}_{1}=\mathscr {A}_{0}^{*}=\mathscr {A}^{*}$ coincides with $\sigma (\mathscr {A})$. This is a case where $\sigma (\mathscr {A})$ can be constructed "from the inside."

Show that the set of normal numbers lies in $\mathscr {I}_{6}$.

Show that $\mathscr {H}^{*}=\mathscr {H}$ if and only if $\mathscr {\mathscr { H }}$ is a $\sigma$-field. Show that $\mathscr {I}_{n-1}$ is strictly smaller than $\mathscr {A}_{n}$ for all $n$.