How compute $\lim_{p\rightarrow 0} \|f\|_p$ in a probability space?

Question

I not solve the follow limit $$\lim_{p\rightarrow 0} \bigg[\int_{\Omega} |f|^p d\mu \bigg]^{1/p} = \exp\bigg[ \int_{\Omega} \log|f|d\mu \bigg],$$

where $(\Omega, \mathcal{F}, \mu)$ is a probability space and $f,\log |f| \in L^1(\Omega).$

Can someone help me?

Thank you!

Answer 1

By Jensen's inequality $$\int \log |f|\le \log \|f\|_q.$$ Hence $$\int \log |f| \le \log \|f\|_q = \frac{1}{q} \log \int |f|^q \le \frac{\int (|f|^q-1)}{q}.$$ Now apply DCT to conclude that the right hand side goes to $\int \log |f|$ as $q$ tends to $0$ from the right.

Thus $$\|f\|_q \to \exp(\int \log |f|) \text{ as } q\to 0^+.$$

Similarly you can work on the left limit.