A sequence $f_k$ defined on the unit ball converging in measure to zero and satisfying $\|f_k\|_{L^2(B(0,1))}\le M$ for all $k\ge 1$.

Question

Let the sequence $f_k$ be defined on the unit ball $B(0,1)=\{x\in\mathbb{R}^d: \|x\|\le 1\}$, converging in measure to zero and satisfying $\|f_k\|_{L^2(B(0,1))}\le M$ for all $k\ge 1$. I would like to show that $f_k\to 0$ in $L^1$.

I know from Cauchy-Schwartz that $$\|f_k\|_{L^1(B(0,1))}\lesssim_d \|f_k\|_{L^2(B(0,1))}.$$

I also know that $\liminf |f_k(x)|^2 \in L^1$ since by Fatou, $$\int \liminf_{k\to\infty} |f_k(x)|^2 \,dx \le \liminf_{k\to\infty} \int |f_k|^2\,dx\le M. $$

Additionally, convergence in measure of $f_k$ implies that there exists a subsequence $f_{n_k}$ such that $f_{n_k}(x)\to 0 $ almost-everywhere. At this point, I would like to show that $f_k\to 0$ in $L^2$. However, I'm not sure how to proceed.

Answer 1

Boundedness of the 2-norms implies uniform integrability. Together with convergence in measure, $L^1$ convergence follows.