Hahn–Kolmogorov theorem

Remarks

Clause 1 uses countable additivity of $\mu_0$: a countable cover of $A \in \mathcal{A}$ by sets of $\mathcal{A}$ can be made disjoint inside $A$, and then its total is at least $\mu_0(A)$. For clause 2, a set $E \in \mathcal{A}$ splits each $A_n$ of a cover additively, so it splits every $B$; the Carathéodory-measurable sets form a $\sigma$-algebra, so they contain $\sigma(\mathcal{A})$. Clause 3 follows from the $\pi$-$\lambda$ theorem, applied on each $X_n$. Example: the finite unions of intervals $(a, b] \cap (0, 1]$ form an algebra on $(0, 1]$, length is a premeasure on it, and the theorem gives Lebesgue measure on the Borel subsets of $(0, 1]$. The measure $\mu^*$ is in fact defined on all Carathéodory-measurable sets, which may be more than $\sigma(\mathcal{A})$; there it is a complete measure.

Going deeper

For a student

This is the algebra form of Carathéodory's extension theorem, with an explicit formula for the extension. In probability it is used to build measures on sequence spaces and, together with a compactness argument, the measures of the Kolmogorov extension theorem. The formula for $\mu^*$ also shows that every measurable set can be approximated in measure by countable unions of sets of $\mathcal{A}$.