Does convergence in every $L^p$ imply convergence in $L^\infty$?

Question

Consider a sequence of functions $(f_n)$ on a probability space $(X,\mu)$ such that

(i) $f_n \to f$ in $L^p$ for every $p\in [1,\infty)$

(ii) $f,f_1,f_2,\cdots\in L^\infty$ and they are uniformly bounded in $L^\infty$ by $\Lambda\in [0,\infty)$, i.e. $$\|f\|_\infty,\|f_1\|_\infty,\|f_2\|_\infty,\cdots\le \Lambda.$$

Does then $f_n\to f$ in $L^\infty$?

score 4 · Accepted Answer · answered Oct 05 '17 at 02:09

4

Let $t(x) = \max(0, 1-|x|)$, let $f_n(x) = t(nx+n)$. Then $f_n \to 0$ for $p < \infty$ but $\|f_n\|_\infty = 1$ for all $n$.

answered Oct 05 '17 at 02:09

copper.hat

172,524

1

Is the $+n$ necessary? If one uses this I think one can get away with $f_n(x)=t(nx)$. – Ryan Unger Oct 05 '17 at 02:32
@0ßelö7: It is not necessary but the pointwise convergence is everywhere instead of ae. – copper.hat Oct 05 '17 at 02:59
Yeah, that's fine. Thanks. – Ryan Unger Oct 05 '17 at 03:04

Does convergence in every $L^p$ imply convergence in $L^\infty$?

1 Answers1