I'm trying to solve an exercise in Rudin about $L^p$ convergence but I need some hint, the exercise is:
let $(X,M,m)$ be a probability space, and let $\Vert{f}\Vert_{r} < \infty$ for some $r$. Prove that
$$\lim_{p -> 0} \Vert{f}\Vert_{p} = \exp\left\{ \int_{X} \log|f| dm \right\}$$ I thought to follow this way: suppose we have proven it for step functions. Let $f$ be as above and let $s_n$ be a sequence of simple functions that monotonically increases to $|f|$. Obviously we have $s_n \in L^p$. Now observe that by the monotone convergence theorem, and some continuity argument we have:
$$ \lim_{n->\infty} \exp\left\{ \int_{X} \log|s_n| dm \right\}=\exp\left\{ \int_{X} \log|f| dm \right\} $$ $$\lim_{n -> \infty} \Vert{s_n}\Vert_{p} = \Vert{f}\Vert_{p} \quad \forall p \le r$$.
Assuming $\exp\{ \int_{X} \log|f|\}$ is finite and using the triangle inequality we get the thesis. If it isn't finite it's trivial by Jensen inequality
The question is: how can i prove it for step functions? I thought to divide it in two cases: given a step function $s = a_1 \mathbb{1}_{A_n} + . . . + a_n \mathbb{1}_{A_n}$ we can consider the case $\sum m(A_i) < 1$ and $\sum m(A_i) = 1$. The first case is not difficult, since it's easy to see that $\exp\left\{\int_{X}\log|s| dm \right\} = 0$.
I can't do the second case, can anyone give me some hint? Thank you