Distribution Function of Finite Borel Measure is Bounded

Theorem

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

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

Then $F_\mu$ is bounded.

Proof

Let $x \in \R$.

Then:

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

Then, from Measure is Monotone we have:

$\map \mu \O \le \map \mu {\hointl {-\infty} x} \le \map \mu \R$

From Empty Set is Null Set, we have:

$\map \mu \O = 0$

Since $\mu$ is a finite measure, we have:

$\map \mu \R < \infty$

So, we have:

$0 \le \map \mu {\hointl {-\infty} x} \le \map \mu \R < \infty$

So:

$0 \le \map {F_\mu} x \le \map \mu \R$ for each $x \in \R$.

$\blacksquare$

Sources

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