This is Tao Analysis II Prop. 19.3.3. (b)
Let $\Omega \subseteq \mathbb R^n$ measurable and $f,g: \Omega \rightarrow \mathbb R$ absolutely integrable functions. Then $f+g$ is absolutely integrable and $$ \int_\Omega f+g = \int_\Omega f + \int_\Omega g $$
How can I prove that ?
My first idea was $f+g = f^+ + g^+ - (f^- + g^-)$. But then I get a problem with "$-$"-sign.