Convergence of $\Vert f \Vert_p$ with respect to $p$ For a Lebesgue measurable function $f$, $\Vert f \Vert_p$ converges to $\Vert f \Vert_\inf$ as $p$ goes to infinity.
I was wondering if $\Vert f \Vert_p$ converges to $\exp(\int \log |f|)$ as $p$ goes to $0$ for probability measure. How to prove this?
Thanks!
 A: As my approach is different to that of Willie Wong, I give it here:
Assume $f$ is in some $L^{p_0}$.
Let $(p_n)$ be such that $0 < p_n < p_0$ and $p_n \downarrow 0$. Define:
$$g_n(x) = \frac{1}{p_0} (|f(x)|^{p_0} - 1) - \frac{1}{p_n} (|f(x)|^{p_n} - 1).$$
Recall that $\frac{1}{p} (x^p - 1) \downarrow \log x$ as $p \downarrow 0$ for all $x > 0$. So let the $g_n \to g$. Then by the Monotone Convergence Theorem we have that
$$\int g_n \uparrow \int g.$$
This implies that 
$$\int \frac{1}{p_n} (|f|^{p_n} -1) \downarrow \int \log |f|.$$
Now, by Jensen's inequality we have
$$\text{exp} \left ( \int \log |f| \right ) \leq \left ( \int |f|^{p_n} \right )^\frac{1}{p_n}$$
and by the inequality $|t| \leq \text{exp}(|t| - 1)$ we get
$$\left ( \int |f|^{p_n} \right )^\frac{1}{p_n} \leq \text{exp} \left ( \int \frac{1}{p_n} (|f|^{p_n} - 1) \right ).$$
So, this together with our previous claim proves the theorem.
A: HINT
Approximate by non-negative step functions (if $f$ vanishes on a set of positive measure, using a Hölder inequality you can argue that $\|f\|_p \leq C |\mathrm{supp}(f)|^{\frac1p - 1} \searrow 0$). For step functions, that $\|f\|_p\geq \exp \int \log |f|$ is a consequence of the arimetic-geometric mean inequality. 
The AM-GM inequality becomes an equality when all its terms are equal. For a non-negative step function $f$, as $p\to 0$, you have that $f \to 1$. By continuity you can quantify the difference between AM-GM. So you have that the convergence is true for step functions. 
To finish off you use the usual diagonal argument to extract a subsequence from the approximating sequence of step functions, together with a suitable sequence of $p$'s. 
