Pre-Measure of Finite Stieltjes Function Extends to Unique Measure

Theorem

Let $\JJ_{ho}$ 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 $\map \BB \R$, the Borel $\sigma$-algebra on $\R$.

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

Proof

We intend to use Carathéodory's Theorem (Measure Theory).

To this end, observe that by Characterization of Euclidean Borel Sigma-Algebra, we have:

$\map \BB \R = \map \sigma {\JJ_{ho} }$

From Pre-Measure of Finite Stieltjes Function is Pre-Measure, $\mu_f$ is a pre-measure.

Note that for all $n \in \N$, we have:

$\map {\mu_f} {\hointr {-n} n} = \map f n - \map f {-n} < +\infty$

and $\hointr {-n} n \uparrow \R$ as an increasing sequence of sets.

All these facts combine with Carathéodory's Theorem (Measure Theory) to establish existence and uniqueness of $\mu$.

$\blacksquare$

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 7$: Problem $9 \ \text{(ii)}$