Pre-Measure of Finite Stieltjes Function is Pre-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 $f$, $\mu_f: \mathcal{J}_{ho} \to \overline \R_{\ge 0}$ is a pre-measure.

Here, $\overline \R_{\ge 0}$ denotes the set of positive extended real numbers.

Proof
It is immediate from the definition of $\mu_f$ that $\mu_f \left({\varnothing}\right) = 0$.

Now suppose that for some half-open interval $\left[{a \,.\,.\, b}\right)$ one has:


 * $\left[{a \,.\,.\, b}\right) = \displaystyle \bigcup_{n \mathop \in \N} \left[{b_n \,.\,.\, b_{n+1}}\right)$

where $b_0 = a$ and $\displaystyle \lim_{n \mathop \to \infty} b_n = b$.

Then we compute:

which verifies the second condition for a pre-measure.

Hence $\mu_f$ is indeed a pre-measure.