This is the popular Bonferroni inequality. Does it also hold for characteristic functions of random variables, as in when $P(A_i)$ is replaced by the characteristic function $\chi(A_i)$ and so forth?
Given random variables $A_1,A_2...A_n$ (not events like in Theorem 1.2.3), is this satisfied? $$ \sum_{i=1}^n||\chi_{A_i}(t)|| - \sum_{i <j}^n||\chi_{A_i|A_j}(t).\chi_{A_j}(t)|| \le ||\chi_{\sum_{i=1}^n A_i}(t)||$$