TITLE: If $\mu(X)=1$, then $\lim_{p\rightarrow 0}\|f\|_p=\exp\left(\int_X\ln(|f|)d\mu\right)$
QUESTION [5 upvotes]: I am trying to prove the following result:

Let $(X,\mu)$ be a measure space of measure 1 and $f$ a complex-valued function on $X$ such that there exists $r>0$ satisfying $\|f\|_r<\infty$. Prove that $\lim_{p\rightarrow 0}\|f\|_p=\exp\left(\int_X\ln(|f|)d\mu\right)$.

I already proved that the RHS term is lower than the LHS term, thanks to Jensen's inequality. However, I find the converse inequality much harder. I thought about defining $X_-=\{x\in X:|f(x)|<1\}$, $X_+=\{x\in X:|f(x)|\geq1\}$ and proving that
$$\forall 0<a<1, \forall b>1, \exists p>0: a\int_{X_-}\ln(|f|)d\mu + b\int_{X_+}\ln(|f|)d\mu\geq \frac{1}{p}\ln\left(\int_X |f|^p d\mu\right),$$
which would imply the result by applying the exponential and considering $a,b\rightarrow 1$ (and recalling that the $p$-norm decreases when $p$ decreases). But I haven't found a way to prove that this last inequality holds for a certain $p$. Is there a way to do it ? (or maybe a simpler way to solve the problem)

REPLY [6 votes]: The proof can be found in Section 6.8 of the old book Inequalities by Hardy, Littlewood and Polya. As Stephen Montgomery-Smith remarks in his answer, the proof is rather easy if $\log |f|$ stays bounded. In the general case, Hardy suggests to write
$$
\int  \log |f| \leq \log \|f\|_q \leq \int \frac{|f|^q-1}{q}
$$
and to remark that, for $t>0$, $q \mapsto \frac{t^q-1}{q}$ decreases to $\log t$ as $q$ decreases to zero. By monotone convergence, you can now let $q \to 0^+$ and conclude.