Pre-Measure of Finite Stieltjes Function is Pre-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 $f$, $\mu_f: \JJ_{ho} \to \overline \R_{\ge 0}$ 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:


 * $\map {\mu_f} \O = 0$

Now suppose that for some half-open interval $\hointr a b$ one has:


 * $\ds \hointr a b = \bigcup_{n \mathop \in \N} \hointr {b_n} {b_{n + 1} }$

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

Then we compute:

which verifies the second condition for a pre-measure.

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