Pre-Measure of Finite Stieltjes Function Extends to Unique Measure

Theorem
Let $\mathcal{J}_{ho}$ denote the collection of half-open intervals in $\R$.

Let $f: \R \to \R$ be a finite Stieltjes function.

Then the pre-measure of $f$, $\mu_f$, extends uniquely to a measure $\mu$ on $\mathcal B \left({\R}\right)$, the Borel $\sigma$-algebra on $\R$.

$\mu$ is the measure of $f$.