Definition 19.4.1 (Elementary Family).label Let $X$ be a set and $\ce \subset 2^{X}$, then $\ce$ is an elementary family if:

1. (P1)

   $\emptyset \in \ce$.

2. (P2)

   For any $A, B \in \ce$, $A \cap B \in \ce$.

3. (E)

   For any $E, F \in \ce$ with $E \subset F$, there exists $\seqf{E_j}\subset \ce$ such that $E \setminus F = \bigsqcup_{j = 1}^{n} E_{j}$.

If $X \in \ce$, then (E) may be replaced with

1. (E’)

   For any $E \in \ce$, there exists $\seqf{E_j}\subset \ce$ such that $E^{c} = \bigsqcup_{j = 1}^{n} E_{j}$.