Set Fields

The pair \((X,\mathcal{F})\) where \(\mathcal{F}\) is a collection of subsets of the set \(X\) is a field of sets if

• \(F\in \mathcal{F} \implies X-F\in \mathcal{F}\)
• \(\emptyset \in \mathcal{F}\)
• \(\forall F_1,\cdots,F_n \in \mathcal{F}\) we have \(\bigcup_{i=1}^n F_i \in \mathcal{F}\) for any \(n\)
• \(\forall F_1,\cdots,F_n\in \mathcal{F}\) we have \(\bigcap_{i=1}^n F_i \in \mathcal{F}\) for any \(n\)

The field \(\mathcal{F}_n\) generated by sets \(X_1,\cdots, X_n\) contains the collection of sets obtained by any sequence of the set operations union, intersection, complements and differences. In this field, the atoms of \(\mathcal{F}_n\) are sets of the form \(\bigcap_{i=1}^n Y_i\) where \(Y_i = X_i\) or \(Y_i=X_i^C\). Atoms are important since any signed measure \(\mu\) on \(\mathcal{F}_n\) is completely specified by the values of \(\mu\) on the atoms.