关于单位球上依测度收敛到0且L²范数有界的序列的L¹收敛性证明问询

Hi folks, I'm stuck on proving the following convergence result and would appreciate some guidance:

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 $L^1$.

Here's what I've worked out so far:

• From the Cauchy-Schwarz inequality, I know that
  $$|f_k|{L^1(B(0,1))}\lesssim_d |f_k|{L^2(B(0,1))}.$$
  (The subscript $d$ means the constant depends on the dimension $d$ of our Euclidean space.)
• Using Fatou's lemma, I've confirmed that $\liminf_{k\to\infty} |f_k(x)|^2$ is in $L^1$, since:
  $$\int \liminf_{k\to\infty} |f_k(x)|^2 ,dx \le \liminf_{k\to\infty} \int |f_k|^2,dx\le M. $$
• I also recall that convergence in measure guarantees the existence of a subsequence $f_{n_k}$ such that $f_{n_k}(x)\to 0$ almost everywhere on $B(0,1)$.

Right now, I'm stuck on bridging these observations to get the $L^1$ convergence. I initially thought proving $f_k\to 0$ in $L^2$ might be a helpful intermediate step, but I can't see how to make that work—or if that's even the right approach.

Any hints, corrections to my current reasoning, or steps to move forward would be really appreciated!

备注：内容来源于stack exchange，提问作者Diffusion