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? $$ \sum_{i=1}^n||\chi_{A_i}(t)|| - \sum_{i <j}^n||\chi_{A_i|A_j}(t).\chi_{A_j}(t)|| \le ||\chi_{\sum_{i=1}^n A_i}(t)||$$
No, the stated Bonferroni style inequality does not hold for characteristic functions. Certainly not in the way you've formulated it, and I can't think of any other way of interpreting the Bonferroni inequality in the context of characteristic functions.
Example: $n=2$, take random variables $X=1_A$ and $Y=1_B$ where $A$ and $B$ are independent events. Then $\chi_X(t)=e^{it}P(A)+P(A^c)$, so we can clearly choose $t$ such that $x:=|\chi_X (t)|<1$. Define also $y:=|\chi_Y(t)|$. In this situation, the stated inequality is equivalent to $x+y-xy\leq xy$ i.e. $x+y\leq 2xy$. Combining this with the AM-GM inequality leads to $\sqrt{xy}\leq xy$, which is clearly false when $xy<1$.
