I'm reading Stein's Real Analysis and I'm looking at P.29, which states $$\textbf{Property 5}. f+g\text{ is measurable if $f,g$ are measurable and finite-valued.}$$ Now, I understand the proof of the statement, which is well-stated in the book and also in the first answer here. However, I'm intrigued as to why the author ensures the condition that $f$ and $g$ remain finite-valued.
Is it simply because $\infty + (-\infty)$ is not defined?
If so, if we are guaranteed that $f,g\geq 0$, then would it be possible for us to safely say that $f+g$ is again measurable if $f$ and $g$ are? I think there is no problem with the proof if the small "problem" of $\infty+-\infty$ doesn't occur, but I would sure like some reassuring opinions from you guys.
Thanks in advance.
EDIT. I've messed around with the proof a bit, and I think my conclusion is that $$\text{$f+g$ is measurable if $f,g$ are measurable and $f+g$ is well-defined a.e.}$$