Absolute Value of Real-Valued Random Variable is Real-Valued Random Variable
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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.
$\blacksquare$