Prove a series finite a.e by proving that its $L_{1,\infty}$ norm is finite.

Question

According to a article, we can show that series $\sum_{i=1}^\infty 2^i\left(\mathbf{1}_{A_i}\right)(x)$ is finite almost surely by proving that its $L_{1,\infty}$ norm is finite. Can you explain me this? With $$\left\| \sum_{i=1}^\infty 2^i(\mathbf{1}_{A_i})(x) \right\|_{1,\infty}= \sup_{\lambda>0} \lambda\cdot\#\left\{x: \sum_{i=1}^\infty 2^i(\mathbf{1}_{A_i})(x) >\lambda\right\}.$$

Answer 1

Assume that we work with the probability space $\left(\Omega,\mathcal F,\mathbb P\right)$. Then the following inequalities hold $$\mathbb P\left(A_i\right)=\mathbb P\left(\left\{2^i\mathbf 1_{A_i}\geqslant 2^i\right\}\right)\leqslant \mathbb P\left(\left\{\sum_{j=1}^{+\infty}2^j\mathbf 1_{A_j}\geqslant 2^i\right\}\right)\leqslant \mathbb P\left(\left\{\sum_{j=1}^{+\infty}2^j\mathbf 1_{A_j}\gt 2^{i-1}\right\}\right)\leqslant 2^{-(i-1)}\left\lVert \sum_{j=1}^{+\infty}2^j\mathbf 1_{A_j}\right\rVert_{1,\infty}. $$ We can conclude by the Borel-Cantelli lemma.