Pre-Measure of Finite Stieltjes Function Extends to Unique Measure

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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 $\R$.

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:

$\mathcal B \left({\R}\right) = \sigma \left({\mathcal{J}_{ho}}\right)$

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:

$\mu_f \left({\left[{-n \,.\,.\, n}\right)}\right) = f \left({n}\right) - f \left({-n}\right) < +\infty$

and $\left[{-n \,.\,.\, n}\right) \uparrow \R$ as an increasing sequence of sets.


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

$\blacksquare$


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