Absolutely Continuous Random Variable is Continuous

Theorem

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

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

Then $X$ is a continuous random variable.

Proof

Let $F_X$ be the cumulative distribution function for $X$.

From the definition of an absolutely continuous random variable, we have:

$F_X$ is absolutely continuous.

From Absolutely Continuous Real Function is Continuous, we then have:

$F_X$ is continuous.

Then, from the definition of a continuous random variable, we have:

$X$ is a continuous random variable.

$\blacksquare$