Measurable functions $f,g$ are finite a.e. Then, $f+g$ is measurable. 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.

The mapping $\phi:(s,t)\mapsto s+t$, from $\Bbb R^2$ to $\Bbb R$ is continuous, hence Borel measurable. On the set $H:=\{x\in\Bbb R^n: f(x)\in\Bbb R, g(x)\in\Bbb R\}$ (given the relative Lebesgue measurable sets), the mapping $\psi:x\mapsto (f(x),g(x))$,  into $\Bbb R^{2}$ with the Borel sets, is  measurable. Therefore the composition $x\mapsto f(x)+g(x) = \phi(\psi(x))$ is Lebesgue measurable on $H$. The assertion follows from the completeness of Lebesgue measure because the complement of $H$ is null.
