Again I am stuck with this problem in Rudin:
Assume that $|f|_{r}<\infty$ for some $r<0$. Prove that $\lim_{p\rightarrow 0}|f|_{p}=e^{\int_{X}\log |f|d\mu}$
Let $r and we have $K=\frac{r}{p}>1$. Denote its conjugate by $K'$. Then we have $\int f^{p}d\mu=\int (f^{p})*1d\mu\le (\int [f^{p}]^{\frac{r}{p}})^{\frac{p}{r}}$ since by assumption $\mu(X)=1$. So in particular we have $|f|_{p}\le |f|_{r}<\infty$ Since $|f|_{p}$ is monotonely decreasing with $p\rightarrow 0$, it must have a limit. We now apply Jensen's inequality, which gives us $\log^{\int_{X}Fd\mu}\ge \int_{X}\log[F]d\mu$ Here $F=f^{p}$. So we have $ \int_{X}f^{p}d\mu\ge (e^{\int_{X}\log[f]d\mu})^{p} $ taking the $p$-th root on both sides we conclude that $|f|_{p}\ge e^{\int \log|f|d\mu}$ But then I got totally stuck. It is worth pointing that Jensen's inequality is only an equality when $f^{p}=c$ is a constant. Therefore $f$ has to be a constant as well.