measurable function by parts

Question

i have the next exercise:

Let $f$ and $g$ be lebesgue measurable functions and $A$ a measurable set. Prove that the function $$h(x)=\left\{\begin{array}{ll}f(x)&\mbox{si }x\in A,\\g(x)&\mbox{ x}\not\in A,\end{array}\right.$$ is measurable.

I'm really lost, and don't know how to start this exercise any help would be appreciated.

Answer 1

If you have already shown that sums and products of measurable functions are measurable, then simply observe that $$ h(x)=f(x)1_A(x)+g(x)1_{A^c}(x) $$ where $1_A$ is the indicator function of the set $A$.

Answer 2

$h(x) \le \alpha$ iff ($x \in A $ and $f(x) \le \alpha$) or ($x \notin A $ and $g(x) \le \alpha$) iff $x \in (A \cap \{ x | f(x) \le \alpha \} \cup (A^c \cap \{ x | g(x) \le \alpha \}$.