# Problem proving \$lim_{n rightarrow infty} int_S(f(x))^{frac{1}{n}}=mu(S)\$

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.