Distribution Function of Finite Borel Measure is Increasing
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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