Statement: Given a measurable set $S \subseteq \mathbb{R}$ and a measurable function $f:S \longrightarrow \mathbb{R}$ with $f(x)\gt1$ for all $x \in S$. Show that
$$\lim_{n \rightarrow \infty} \int_S(f(x))^{\frac{1}{n}}=\mu(S)$$
I have searched that there are some similar questions here and here, but they do not start from the same hyphothesis.The sequence $f_n = (f(x))^{\frac{1}{n}}$ is decreasing and I have tried to solve it by dominated/monotone convergence, but i believe that there is something missing about the integrability of $f$. I do not know how to continue.