Positive Part 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 the positive part $X^+$ of $X$ is a real-valued random variable.

Proof

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

From Function Measurable iff Positive and Negative Parts Measurable, $X^+$ is $\Sigma$-measurable.

So $X^+$ is a real-valued random variable.

$\blacksquare$