Theorem

combo extension

Suppose $\mathcal{S}$ is a semialgebra of subsets of $\Omega$ and that $\mathbb{P}: \mathcal{S} \to [0, 1]$ is $\sigma$-additive with $\mathbb{P}(\Omega) = 1$. Then there is a unique probability measure on $\sigma(\mathcal{S})$ that extends $\mathbb{P}$.