Let $\{f_n\}^∞_{n=1} ⊂ L_2(μ)$ be a sequence of functions such that $∥f_n∥_2 ≤ 1.$

Question

Let $(X,A,μ)$ be a finite measure space. Let $\{f_n\}^∞_{n=1} ⊂ L_2(μ)$ be a sequence of functions such that $∥f_n∥_2 ≤ 1.$

a.) Prove that if $f_n → 0$ in measure, then $f_n → 0$ in $L_1(μ).$

b.) If $f_n → 0$ in measure, does it necessarily follow that $f_n → 0$ in $L_2(μ)?$

Not sure how to begin this problem. Its from a past qual. In particular I am at a loss at what to do with the $L_2$ condition. Thanks for any help.

Answer 1

For b), consider the following variant of the Vitali theorem. If $(X,A,\mu)$ is a finite measure space, then the statements "$f_n^p$ is uniformly integrable and $f_n \to f$ in measure" and "$f_n \to f$ in $L^p$" are equivalent. So you want to see whether these conditions imply that $f_n^2$ is uniformly integrable. If they do, prove it. If they don't, try and produce an example which is not uniformly integrable and also satisfies the boundedness hypothesis.

A more specific hint: if your condition were boundedness in $L^1$ instead of $L^2$, then $f_n(x) = n \chi_{[0,1/n]}$ would be a counterexample, since it converges in measure to 0 but the integrals are all 1. Can you adapt this to the setting of $L^2$?