$\lim_{p \rightarrow 0} ||f||_p=e^{\int_0^1 ln|f|}$

Question

Let $f\in L^p[0,1]$ for all $p>0$ show that $$\lim_{p \rightarrow 0} ||f||_p=e^{\int_0^1 ln|f|}$$ If we write $|f|=e^{ln |f|}$ then $|f|^p=e^{p ln|f|}$. so $$||f||_p^p=\int_0^1 e^{p ln |f|}$$

Answer 1

Hint:

Using the fact that $\ln$ is a concave function and Jensen's inequality,

$$\int_0^1\ln |f| \leqslant\frac{1}{p}\ln \left(\int_0^1 |f|^p \right)\leqslant \frac{1}{p} \left( \int_0^1 |f|^p - 1 \right)= \int_0^1 \left(\frac{|f|^p -1}{p}\right)$$

Apply the monotone convergence theorem to find the limit of the RHS.