Measure of Stieltjes Function of Measure

Theorem
Let $\mu$ be a measure on $\map \BB \R$, the Borel $\sigma$-algebra on $\R$.

Suppose that for all $n \in \N$, $\mu$ satisfies:


 * $\map \mu {\hointr {-n} n} < +\infty$

Let $f_\mu$ be the Stieltjes function of $\mu$.

Let $\mu_{f_\mu}$ be the measure of $f_\mu$.

Then $\mu_{f_\mu} = \mu$.

Proof
From Pre-Measure of Finite Stieltjes Function Extends to Unique Measure, it suffices to verify that:


 * $\map {\mu_{f_\mu} } {\hointr a b} = \map \mu {\hointr a b}$

for all half-open intervals $\hointr a b$.

Now we have:

If either $a = 0$ or $b = 0$, the result follows immediately from the definition of $f_\mu$.

Now suppose that $a < b < 0$.

Then:

Finally, let $0 < a < b$.

Then:

The final case $a < 0 < b$ is a trivial consequence of Measure is Finitely Additive Function.

Hence it must be that $\mu_{f_\mu} = \mu$.