Discrete probability spaces revisited

TL;DR

The paper addresses the problem of extending a probability measure defined on an arbitrary $\sigma$-field $\mathcal{F}$ on a countable $\Omega$ to a probability measure on the full power set $\mathcal{P}(\Omega)$. It provides an elementary, constructive proof of this extension by first obtaining a countable $\mathcal{F}$-measurable partition of $\Omega$ and then defining a local probability mass function on each block, finite blocks assigned uniformly and infinite blocks assigned a decaying scheme via a bijection to $\mathbb{N}$. This yields a global pmf $p$ with $\mathbb{P}(A)=\sum_{\omega\in A}p(\omega)$ for all $A\in\mathcal{F}$, demonstrating the existence of an extension $\mathbb{P}^*$ to $\mathcal{P}(\Omega)$; the construction also shows a one-to-one correspondence between blockwise pmfs and extensions. The result, connected to classical work by Bierlein and Bogachev, provides an accessible proof suitable for introductory probability courses and clarifies how discrete random variables on countable spaces can have full distribution information recovered on the power set.

Abstract

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

Theorems & Definitions (2)

• proof
• proof : Proof of Theorem \ref{['maintheo']}