Let $E\in \mathbb{R}^n$ be a Lebesgue measurable set.
Let $f,g:\mathbb{R}^n \to \overline{\mathbb{R}}$ be Lebesgue measurable functions. Suppose $f$ and $g$ are finite almost everywhere. Then, prove that $f+g$ is a Lebesgue measurable function.
In $f+g$ is measurable no matter how it is defined at points where it has the form $\infty-\infty.$?, use the fact that if $f$ is measurable and $f=g$ a.e., then $g$ is measurable.
But I don't know how I should use the fact.
Thank you for your help. Other way to prove is also welcomed.
Best Answer
Suppose $f,g: \mathbb R^n \to \mathbb R\cup\{\pm\infty\}.$
Let $h(x) = \begin{cases} g(x) & \text{if } g(x)\in\mathbb R, \\ 0 & \text{if } g(x)\in\{\pm\infty\}. \end{cases}$
Now use the fact that if $g$ is measurable and $g=h$ a.e., then $h$ is measurable.
You have $f+g= f+h$ a.e. So the problem of showing $f+g$ is measurable is reduced to that of showing $f+h$ is measurable, and here you have no $\infty-\infty$ problem.