Lemma

Let $\Phi$ be a nonempty set of subsets of $A$ satisfying the axiom (CL). Given a finite family $(M_i)_{1 \leq i \leq n}$ of sets in $\Phi$, there exists a

¹ A clan $\Phi$ of subsets of a set $A$ is also known as a ring (or Boolean ring) of sets; if, moreover, $A \in \Phi$, then $\Phi$ is called an algebra (or Boolean algebra) of sets (cf. GT, I, §6, Exer. 20 and II, §4, Exer. 12). A Boolean algebra closed under countable unions is called a tribe, or $\sigma$-*algebra* (GT, IX, §6, No. 3, Def. 3).

finite family $(N_j)_{1 \leq j \leq m}$ of pairwise disjoint sets in $\Phi$ such that each $M_i$ is the union of a certain number of the $N_j$.

For, consider the $2^n - 1$ sets of the form $\bigcap_{i=1}^n P_i$, where $P_i = M_i$ for certain indices $i$, $P_i = \mathbf{C} M_i$ for the others, at least one of the $P_i$ being equal to $M_i$. Let $(N_j)_{1 \leq j \leq m}$ be the sequence of these sets arranged in some order; they are pairwise disjoint and belong to $\Phi$; on the other hand, every set $M_k$ is the union of the sets $N_j = \bigcap_{i=1}^n P_i$ corresponding to the families $(P_i)$ such that $P_k = M_k$, which proves the lemma.

The lemma established, every function of the form $\sum_{i=1}^n c_i \varphi_{M_i}$, where $M_i \in \Phi$, may be written in the form $\sum_{j=1}^m d_j \varphi_{N_j}$, where the $N_j$ belong to $\Phi$ and are pairwise disjoint; if $\varphi_M = \sum_{j=1}^m d_j \varphi_{N_j}$ then necessarily $d_j = 0$ or $d_j = 1$ for each index $j$, thus $M$ is the union of a certain number of the $N_j$, and therefore belongs to $\Phi$.

Every clan $\Phi$ of subsets of $A$ contains the empty subset $\varnothing$ of $A$; for, there exists at least one subset $M \in \Phi$, therefore $M - M = \varnothing$ belongs to $\Phi$. Note also that the set of subsets of $A$ consisting of the single subset $\varnothing$ is a clan.