Suppose the following condition holds : $$\sum_k \mathbb{E}[\| w_k \|^2 ] < \infty,$$ where $w_k$ is some sequence and $\mathbb{E}[\cdot]$ is the expectation. I want to show that $\sum_k \|w_k\|^2 < \infty$
Is the following reasoning true : $$ \sum_k \mathbb{E}[\| w_k \|^2 ] = \mathbb{E}[\sum_k \| w_k \|^2 ] $$ by monotone convergence theorem. Since this is finite , we have almost surely $\sum_k \|w_k\|^2 < \infty$.