Distribution Function of Finite Borel Measure is Increasing

Theorem

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

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

Then $F_\mu$ is an increasing function.

Proof

Let $x, y \in \R$ be such that $x \le y$.

Then:

$\hointl {-\infty} x \subseteq \hointl {-\infty} x$

So, from Measure is Monotone:

$\map \mu {\hointl {-\infty} x} \le \map \mu {\hointl {-\infty} y}$

That is:

$\map {F_\mu} x \le \map {F_\mu} y$ whenever $x \le y$.

So $F_\mu$ is an increasing function.

$\blacksquare$

Sources

• 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $1.3$: Outer Measures