A measurable function is defined the following way:
$f:X \rightarrow [-\infty, +\infty]$ such that $f^{-1}((-\infty, a))\in \mathcal{M} \ \ \forall a \in \mathbb{R}$
Then, there is this following problem:
Given $E$ a measurable set, $f,g:E \rightarrow \mathbb{R}$ such that $f(x), g(x) \geq 1 \ \ \forall x \in \mathbb{R}$ measurable funcitions then $h:E\rightarrow \mathbb{R}$ defined $h(x):=f(x)^{g(x)}$ is measurable.
I've tried applying the definition directly but i fail to see how to solve the problem. I would appreciate some help or hints on the proof.
Thanks in advance.