Whether or not a certain function is measurable Let $(X,\mathscr{A},\mu)$ be a measure space, let $f:X\rightarrow[0,\infty)$ be measurable, and let $u_n:X\rightarrow(0,\infty)$ be measurable for each $n\in\mathbb{N}$. I want to know if $\left( 1 + \frac{f}{u_n}\right)^{u_n}$ is measurable.
I need this step to prove the convergence of a certain sequence of integrals.
I know that $\left( 1 + \frac{f}{u_n}\right)$ is measurable since $u_n$ is never zero. I just don't know how to show it is measurable when I raise it to $u_n$.
Any ideas? 
Thanks.

Hint: Rather than looking at the function as
$$
\left( 1 + \frac{f}{u_n}\right)^{u_n},
$$
how about 
$$
\exp \left( u_n \log \left( 1 + \frac{f}{u_n} \right)\right)?
$$
