Caratheodory’s Extension

Motivation

To define a (probability) measure on a general Sigma Field $\mathcal{F}$, we start with a smaller collection, ${\mathcal{F}}_0\subset \mathcal{F}$, of subsets of Ω, which is a field (closed under finite union), and on which the desired probabilities are easy to define. Furthermore, we make sure that ${\mathcal{F}}_0$ is rich enough, so that the $\sigma$-field it generates is the same as the desired $\sigma$-field $\mathcal{F}$. We then extend the definition of the probability measure from ${\mathcal{F}}_0$ to the $\mathcal{F}$.

Cathéodory's Extension Theorem

Let ${\mathcal{F}}_0$ be a field of subsets of $\Omega$ and let $\mathcal{F}=\sigma ({\mathcal{F}}_0)$ be the $\sigma$-field it generates. Suppose ${\mathbb{P}}_0$ is a mapping from ${\mathcal{F}}_0$ to $[0,1]$ such that ${\mathbb{P}}_0(\Omega )=1$ and is countably additive. Then, ${\mathbb{P}}_0$ can be extended uniquely to a probability measure $\mathbb{P}$ on $\mathcal{F}$. That is, there exists a unique probability measure $\mathbb{P}$ on $(\Omega ,\mathcal{F})$ such that $\mathbb{P}(A)={\mathbb{P}}_0(A)$ for all $A\in {\mathcal{F}}_0$.

Remarks

• The main hurdle in applying the extension theorem is the verification of the countable additivity property of ${\mathbb{P}}_0$ on ${\mathcal{F}}_0$.
• Alternatively, by Continuity of Probability Measures, it suffices to verify that if $\left\{A_i\right\}$ is a decreasing sequence of sets in ${\mathcal{F}}_0$ and if ${\cap }_{i=1}^{\infty }{A}_i$ is empty, then ${\lim }_{n\to \infty }{\mathbb{P}}_0(A_i)=0$.