Theorem

first measure extension

Suppose $\mathcal{S}$ is a semialgebra of subsets of $\Omega$ and $\mathbb{P}: \mathcal{S} \to [0, 1]$ is $\sigma$-additive on $\mathcal{S}$ and satisfies $\mathbb{P}(\Omega) = 1$. Then there is a unique extension $\mathbb{P}'$ of $\mathbb{P}$ to $\mathcal{A}(\mathcal{S})$, the field generated by $\mathcal{S}$, defined by $\mathbb{P}'(\Sigma_{i \in I}S_i) = \Sigma_{i \in I}\mathbb{P}(S_i)$, which is a probability measure on $\mathcal{A}(\mathcal{S})$; i.e. $\mathbb{P}'(\Omega) = 1$ and $\mathbb{P}'$ is $\sigma$-additive on $\mathcal{A}(\mathcal{S})$.