# the inclusion-exclusion principle for an infinite indexed family of sets

Assume that $\{A_{i}\}_{i\in I}$ is a collection of non-empty sets indexed by an arbitrary indexing set $I$. How can I construct a pairwise disjoint collection $\{A'_{i}\}_{i\in I}$ such that $A'_{i}\subset A_{i}$ for every $i$, and $${\bigcup}_{i\in I}A_{i}'={\bigcup}_{i\in I}A_{i}?$$ (By pairwise disjoint, I mean, of course, that $i\neq j$ implies $A_{i}'\cap A_{j}' = \emptyset.$) If $I$ were finite, I could use the inclusion-exclusion principle. But would this work when $I$ is countably infinite, or when $I$ is uncountable?

Assuming $$I$$ is truly arbitrary, I think you would need choice. Well order $$I$$ with order type $$\alpha$$, and define sets $$A'_\beta$$, $$\beta < \alpha$$ by $$A'_\beta = A_\beta \setminus \bigcup_{\gamma<\beta} A_\gamma$$
The $$A_\beta$$ are pairwise disjoint by construction, and their union is still the same: for every $$x \in \bigcup A_i$$, there is unique $$\beta$$ with $$x \in A'_\beta$$; in fact, $$\beta = \min \{ \gamma : x \in A_\gamma \}$$.