Consider a measurable space $(\Omega, \mathscr{F}, P)$ with $P(\Omega) = 1$. Define for measurable functions $X$ the following $\| X \|_p := \left(\int |X|^p dP\right)^{1/p}$. We know that for $p \in [1, \infty)$ that this is a norm, the $L^p$ norm. Let $S = \{p : 0 < p < \infty \text{ and } \|X \|_p <\infty \}$ and assume that $S \neq \emptyset$. Prove $$ \lim_{p \downarrow 0} \| X \|_p = \exp \{ \int \log |X| \; dP \} $$ defining $\exp\{ - \infty \} := 0$.
Disclaimer: This was a homework problem in a graduate measure theory. The teacher never distributed solutions, and I have struggled to prove it since I first saw it.
