Function Measurable iff Positive and Negative Parts Measurable

Theorem
Let $\left({X, \Sigma}\right)$ be a measurable space.

Let $f: X \to \overline{\R}$ be an extended real-valued function.

Let $f^+, f^-: X \to \overline{\R}$ be the positive and negative parts of $f$.

Then $f$ is $\Sigma$-measurable iff both $f^+$ and $f^-$ are $\Sigma$-measurable.