Measure of Half-Open Interval as Difference of Distribution Function

Theorem

Let $a, b \in \R$.

Let $\mu$ be a finite Borel measure on $\R$.

Let $F_\mu$ be the distribution function of $\mu$.

Then:

$\map \mu {\hointl a b} = \map {F_\mu} b - \map {F_\mu} a$

Note that:

$\map \mu {\hointl {-\infty} a} < \infty$

since $\mu$ is finite.

Then, we have:

$\ds \map {F_\mu} b - \map {F_\mu} a$

$=$

$\ds \map \mu {\hointl {-\infty} b} - \map \mu {\hointl {-\infty} a}$

$\ds $

$=$

$\ds \map \mu {\hointl {-\infty} b \setminus \hointl {-\infty} a}$

$\ds $

$=$

$\ds \map \mu {\hointl a b}$

$\blacksquare$