Showing that a A measurable function is integrable over the union of two sets iff its integrable over each set individually.

Question

Im trying to show that if we have a measure space $(X,\mathcal{A},\mu)$ and $f:X\rightarrow{\mathbb{R}}$ an $\mathcal{A}$-measurable function and $A,B\in{\mathcal{A}}$ two disjoint sets, then $f$ is integrable over $A\cup{B}$ iff $f$ is integrable over $A$ and over $B$.

Ive been toying around with the definitions but can't really seem to get anywhere. I understand the intuition.

Answer 1

Hint: $\int_{A\cup B} |f| \; d\mu = \int_A |f|\; d\mu + \int_B |f|\; d\mu$.