Cumulative Distribution Function is Increasing

Theorem

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $F_X$ be the cumulative distribution function.

Then:

$F_X$ is an increasing function.

Proof

Let $x, y \in \R$ have $x \le y$.

Note that if $\omega \in \Omega$ is such that:

$\map X \omega \le x$

then:

$\map X \omega \le y$

so:

$\set {\omega \in \Omega : \map X \omega \le x} \subseteq \set {\omega \in \Omega : \map X \omega \le y}$

From Measure is Monotone, we then have:

$\map \Pr {X \le x} \le \map \Pr {X \le y}$

That is, from the definition of cumulative distribution function, we have:

$\map {F_X} x \le \map {F_X} y$

whenever $x \le y$.

So:

$F_X$ is an increasing function.

$\blacksquare$