Function that converges in $L^\infty$ but not in $L^p$

Question

I know that convergence in $L^\infty$ implies convergence in $L^p$ in finite measure spaces since if it is finite then Holder's inequality will hold and this can be proven.

I am trying to think of a function that converges in $L^\infty$ but does not converge in $L^p$ for some $p$ or maybe even for all $p$?

Answer 1

Find a sequence of sets $A_n$ with measure going to infinity fast enough so that the functions $$ f_n = \frac{1}{n}\;\mathbf1_{A_n} $$ satisfy $\|f_n\|_\infty \to 0$ but $\|f_n\|_p \to \infty$ for all $p$.

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added
Let $A_n$ have measure $2^n$. Then $$ \|f\|_\infty = \frac{1}{n} \to 0, \\ \|f\|_p = \left(\int_{A_n} \frac{1}{n^p}\right)^{1/p} = \frac{2^{n/p}}{n} \to \infty . $$

Answer 2

Take $f_n=\frac{1}{n}1_{[0,n]}$. It converges in $L^{\infty}$ but not in $L^1$.