the relationship between $L^p$'s norm and $L^\infty$ norm

Question

suppose $f\in L^p(X,\mathscr{A},\mu), \forall p\in[1,\infty)$, then \begin{equation} \| f\|_{\infty}=\lim_{p\rightarrow\infty}\| f\|_{p} \end{equation}

I can prove it on condition that $X$ is finite measure, and I think if not, it is not true, but I do not have a counterexample.

thanks