I'm reading https://faculty.math.illinois.edu/~psdey/MATH561SP19/week7.pdf for a short proof on the Kolmogorov's inequality but I don't understand this line
Then $$E(S_{n}^{2}) \geq \sum_{i=1}^{n} E(S_{n}^{2} \mathbb{1}_{A_i}) = \dots$$
where $1_{A_i}$ is the indicator function on $A_i$. How can I interpret this first inequality? Thanks
Note that $\ A_i\cap A_j=\emptyset\ $ for $\ i\ne j\ $, as your source points out in the paragraph immediately preceding the inequality. Therefore, $\ \displaystyle\sum_{i=1}^n\mathbb{1}_{A_i}= \mathbb{1}_{\bigcup_\limits{i=1}^nA_i}\le1\ $ with certainty, and since $\ S_n^2\ge0\ $ with certainty, then $$ \sum_{i=1}^n E\left(S_n^2 \mathbb{1}_{A_i}\right)=E \left(S_n^2 \sum_{i=1}^n \mathbb{1}_{A_i}\right)\le E\left(S_n^2\right)\ . $$