Weighted inclusion-exclusion principle (${\sum }_{x\notin {\bigcup }_i{A}_i}f(x)={\sum }_{S\subseteq \{1,\dots ,m\}}(-1{)}^{|S|}{\sum }_{x\in {\bigcap }_{i\in S}{A}_i}f(x)$)

For finite subsets $A_i$ of a finite ambient set $X$, the indicator function of their complement is ${\prod }_i(1-1_{A_i})$. Expanding this finite product and summing it against any weight $f:X\to \mathbb{R}$ proves the weighted formula. The empty intersection is $X$. Negative weights are allowed; finiteness removes every convergence issue.