Countably Additive Probability

Show that if $P$ is countably additive, then it is also finitely additive; that is, $\mathbb {P}\left(A \cup B\right)=\mathbb {P}\left(A\right)+\mathbb {P}\left(B\right)$ for any $A, B \in \mathcal{F}$ such that $A \cap B =\varnothing$.

Let $\mathcal{F}$ be a finite field. Show that if $P$ is a finitely additive probability measure on $\mathcal{F}$, then it is also countably additive.

Let $\Omega =\mathbb {N}$ and let $\mathcal{F}$ be the family of all subsets of $\Omega$. Can $\mathbb {P}\left(\{ i\} \right)= \frac{1}{i}, i=1,2, \ldots$, be extended to a probability measure on $\mathcal{F}$?

Let $\Omega =\mathbb {N}$, and let $\mathcal{F}$ be the family of all subsets of $\Omega$. Put $\mathbb {P}\left(\{ i\} \right)= \alpha_{i}, i=1,2, \ldots$. Extend $P$ to a probability measure defined on $\mathcal{F}$. What conditions have to be imposed on the numbers $\alpha_{i}$? Can they all be chosen the same?

Let $\Omega =\mathbb {N}$ and let $\mathcal{F}$ be the family of all subsets of $\Omega$. Can $\mathbb {P}\left(\{ i\} \right)= 2^{-i}, i=1,2, \ldots$, be extended to a probability measure on $\mathcal{F}$?

Let $\Omega =\mathbb {N}$ and let $\mathcal{F}$ be the family of all subsets of $\Omega$. Can you find a value of $c$ such that $\mathbb {P}\left(\{ i\} \right)=\frac{c}{i!}, i=1,2, \ldots$, can be extended to a probability measure on $\mathcal{F}$?

Let $\Omega =\mathbb {N}$ and let $\mathcal{F}$ be the family of all subsets of $\Omega$. Is

a probability measure on $\mathcal{F}$?

Show that if $A_{1} \subset A_{2} \subset A_{3} \subset \cdots$ (this is called an expanding or increasing sequence of sets), then

Show that if $A_{1} \supset A_{2} \supset A_{3} \supset \cdots$ (this is called a contracting or decreasing sequence of sets), then

If $\mathbb {P}\left(A_{n}\right)=0$ for $n=1,2, \ldots$, then $\mathbb {P}\left(\bigcup_{n=1}^{\infty } A_{n}\right)=0$.

If $\mathbb {P}\left(A_{n}\right)=1$ for $n=1,2, \ldots$, then $\mathbb {P}\left(\bigcap_{n=1}^{\infty } A_{n}\right)=1$.

If $\mathbb {P}\left(A_{n} \triangle A\right) \rightarrow 0$ as $n \rightarrow \infty$, then $\mathbb {P}\left(A_{n}\right) \rightarrow \mathbb {P}\left(A\right)$ as $n \rightarrow \infty$.

Prove Theorem 6.1. (It is called the first Borel-Cantelli lemma.)

Verify that

Find $m\left(\left[\frac{1}{2}, \frac{2}{3}\right)\right), m\left(\left(\frac{1}{5}, \frac{1}{3}\right]\right), m(\{ 1\} ), m\left(\left\{ \frac{1}{n}: n=1,2, \ldots \right\} \right)$, and $m(\mathbb {N})$.

Find $m\left(\left[\frac{1}{4}, \frac{1}{2}\right]\right), m\left(\left\{ 0, \frac{1}{2}, 1\right\} \right)$, and $m(\mathbb {Q} \cap [0,1])$ ($\mathbb {Q}$ denotes the set of all rational numbers).

Find $m\left(\left\{ x:\left|x-n\right|<\frac{1}{2^{n}} \text{ for some } n \in \mathbb {N}\right\} \right)$.

Find $m\left(\bigcup_{n=1}^{\infty }\left(\frac{1}{2^{n+1}}, \frac{1}{2^{n}}\right]\right)$.

Find $m(C)$, where $C$ is the Cantor set (see Problem 5.23 for the definition of the Cantor set).

$f(x)=c \; \mathbb {1}_{[1,3]}(x)$.

$f(x)=c \; \mathbb {1}_{[a, b]}(x)$.

$f(x)=c x^{n}$ for $x \in (0,1)$ and zero otherwise, where $n \in \mathbb {N}$.

$f(x)=c e^{-\lambda x}$ for $x>0$ and zero otherwise.

$f(x)=c x(1-x)$ for $x \in [0,1]$ and zero otherwise.

$f(x)=c(x-[x])$ for $x \in [0,2]$ and zero otherwise ($[x]$ denotes the largest integer not greater than $x$).

$f(x)=x^{2} \; \mathbb {1}_{[0, c]}(x)$.

Prove Theorem 6.2.

$f(x)=\frac{1}{2} \; \mathbb {1}_{[1,3]}(x)$, $A=[0,2)$.

$f(x)=x-[x]$ for $x \in [0,2]$ and zero otherwise, $A=\left[\frac{1}{2}, 2\right]$ ($[x]$ denotes the largest integer not greater than $x$).

$f(x)=e^{-x}$ for $x \geqslant 0$ and zero otherwise, $A=\mathbb {N}$.

$f(x)=\frac{1}{5} \; \mathbb {1}_{[-2,3]}(x)$, $A=(0,1)$.

$f(x)=\frac{1}{2} x^{-\frac{1}{2}}$ for $x \in (0,1)$ and zero otherwise, $A=\left(0, \frac{1}{2}\right)$.

Show that $\delta_{\omega }$ is a probability measure. What is the largest set of measure 0? What is the smallest set of measure 1?

Let $\omega_{1}, \omega_{2} \in \Omega$. Show that $\mathbb {P}\left(A\right)=\frac{1}{2} \delta_{\omega_{1}}(A)+\frac{1}{2} \delta_{\omega_{2}}(A)$ is a probability measure. Which sets have measure 0?

What are the conditions on $\alpha_{k}$ for $P=\sum_{k=1}^{\infty } \alpha_{k} \delta_{\omega_{k}}$ to be a probability measure? Which sets have measure 0?

If $P_{1}$ and $P_{2}$ are probability measures, then $\mathbb {P}\left(A\right)=\alpha_{1} \mathbb {P}_{1}\left(A\right)+ \alpha_{2} \mathbb {P}_{2}\left(A\right)$ is also a probability measure provided that $\alpha_{1}$ and $\alpha_{2}$ are non-negative and $\alpha_{1}+\alpha_{2}=1$.

If $P_{k}, k=1,2, \ldots$, are probability measures, $\alpha_{k} \geqslant 0$ for all $k$, and $\sum_{k=1}^{\infty } \alpha_{k}=1$, then $\mathbb {P}\left(A\right)=\sum_{k=1}^{\infty } \alpha_{k} \mathbb {P}_{k}\left(A\right)$ is a probability measure.

Define $P$ by $\mathbb {P}\left(A\right)=\frac{1}{3} \delta_{0}(A)+\frac{2}{3} \mathbb {P}_{2}\left(A\right)$, where $P_{2}$ has density $f(x)= \frac{1}{2} \; \mathbb {1}_{[1,3]}(x)$. Compute $\mathbb {P}\left([0,2]\right)$.

Define $P$ by $\mathbb {P}\left(A\right)=\frac{1}{8} \delta_{1}(A)+\frac{3}{8} \delta_{2}(A)+\frac{1}{2} \mathbb {P}_{3}\left(A\right)$, where $P_{3}$ is given by the density $f(x)=\frac{1}{2} \; \mathbb {1}_{[1,3]}(x)$. Compute $\mathbb {P}\left([2,3]\right)$.

Define $P$ by $\mathbb {P}\left(A\right)=\frac{1}{4} \delta_{0}(A)+\frac{1}{2} \delta_{2}(A)+\frac{1}{4} \mathbb {P}_{3}\left(A\right)$, where $P_{3}$ has density $f(x)=e^{-x} \; \mathbb {1}_{[0,+\infty )}(x)$. Compute $\mathbb {P}\left([0,1]\right)$.

Show that if $X$ is an uncountable set of positive real numbers, then there is a sequence $x_{1}, x_{2}, \ldots \in X$ such that $\sum_{i=1}^{\infty } x_{i}=\infty$.