If $f \in L^{p\_0}(X,M,\,u)$ for some $0<p\_0 \le\infty$, then
$$1. \lim\_{p\to0}\int\_X |f|^p \, d\mu=\mu(\{x \in X \mid f(x) \ne0\}).$$

And if additional assume $\mu(X)=1$,

then I wanna prove that $f \in L^p(X,M,\,u)$ for some $0<p \le p\_0$, and the equation below.
$$2. \lim\_{p\to0}\|f\|\_p=e^{\int\_X\log|f|\,d\mu}$$

I want to know how can I conclude those results. First, I'm trying to use integrate over the set on which $0<|f(x)|\le1$ and use MCT. and the set on which $|f(x)|>1$ use LDCT to prove that. But I can't conclude to measure $\mu$ and how can I approach second fact?