Pointwise Sum of Measurable Functions is Measurable

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

Let $f, g: X \to \overline{\R}$ be $\Sigma$-measurable functions.

Assume that the pointwise sum $f + g: X \to \overline{\R}$ is well-defined.

Then $f + g$ is a $\Sigma$-measurable function.