# Absolute Value of Real-Valued Random Variable is Real-Valued Random Variable

## Theorem

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

Let $X$ be a real-valued random variable.

Then $\size X$ is a real-valued random variable.

## Proof

Since $X$ is a real-valued random variable, $X$ is $\Sigma$-measurable.

From Absolute Value of Measurable Function is Measurable, $\size X$ is $\Sigma$-measurable.

So $\size X$ is a real-valued random variable.

$\blacksquare$