Converge in $L^p$ implies convergence in $L^q$

Question

Let $(E,M,\mu)$ be a measured space. Let $p,q \in[1,\infty]$ with $p\neq q$, and let $(f_n)$ be a sequence of functions in $L^p\cap L^q$. Assume that $f_n\to 0 $ as $n \to \infty$ and $（f_n）$ is Cauchy in $L^q$. How can we show $f_n\to 0$ in $L^q$?

If it's Cauchy in $L^q$, it converges to something in $L^q$. Now deduce that that something must be $0$. — Daniel Fischer, Oct 23 '13 at 22:45

score 6 · Answer 1 · answered Oct 23 '13 at 23:40

6

For $p<q$, convergence in $L_{q}$ implies convergence in $L_{p}$ (Hint: Prove this with Jensen's Inequality)

answered Oct 23 '13 at 23:40

Mael

741

14

Only when the space has finite measure. – Daniel Fischer Oct 23 '13 at 23:45

Converge in $L^p$ implies convergence in $L^q$

1 Answers1