Existence of Probability Space and Discrete Random Variable

Theorem

Let $I$ be some indexing set.

Let $S = \left\{{s_i: i \in I}\right\} \subset \R$ be a countable set of real numbers.

Let $\left\{{\pi_i: i \in I}\right\} \subset \R$ be a countable set of real numbers which satisfies:

$\displaystyle \forall i \in I: \pi_i \ge 0, \sum_{i \in I} \pi_i = 1$

Then there exists a probability space $\left({\Omega, \Sigma, \Pr}\right)$ and a discrete random variable $X$ on $\left({\Omega, \Sigma, \Pr}\right)$ such that the probability mass function $p_X$ of $X$ is given by:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle p_X \left({s_i}\right)\)	\(=\)	\(\displaystyle \pi_i\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	if $i \in I$
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle p_X \left({s}\right)\)	\(=\)	\(\displaystyle 0\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	if $s \notin S$

Proof

Take $\Omega = S$ and $\Sigma = \mathcal P \left({S}\right)$.

Then let:

$\displaystyle \Pr \left({A}\right) = \sum_{i: s_i \in A} \pi_i$

for all $A \in \Sigma$.

Then we can define $X: \Omega \to \R$ by:

$\forall \omega \in \Omega: X \left({\omega}\right) = \omega$

This suits the conditions of the assertion well enough.

$\blacksquare$

Comment

What this theorem allows us to do is ignore all the detail of sample spaces, event spaces and probability measure, and merely say:

"For each $i \in I$, let $X$ be a random variable which takes value $s_i$ with probability $\pi_i$"

and we know that such a random variable exists without having construct it every time.

Sources

• Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction (1986): $\S 2.1$: Theorem $2 \text{A}$