Discrete probability spaces revisited

TL;DR

This paper provides an elementary proof that any probability measure on a countable sample space can be extended to the entire power set, with discussions on its implications for discrete random variables.

Contribution

It offers a simple proof of measure extension on countable spaces and explores its relevance to discrete random variables.

Findings

Every probability measure on a countable space extends to the power set.

The extension is elementary and accessible.

Implications for discrete random variables are discussed.

Abstract

We give an elementary proof of the known fact that every probability measure, defined on an arbitrary $\sigma$-field on a countable sample space $\Omega$, may in fact be extended to a probability measure on the power set of $\Omega$. This result is further discussed and motivated in the context of discrete random variables.