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 $$\left\{{\pi_i: i \in I}\right\}$$ which satisfies:
 * $$\forall i \in I: \pi_i \ge 0, \sum_{i \in I} \pi_i = 1$$

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:

$$ $$

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

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

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.