Lemma 2.2.697A1ECFF2340

Proof. (1) Let $\mathcal L$ be the intersection of all families of subsets of $Z$ which contain $\mathcal P$ and are closed under complements and countable disjoint unions; it is the smallest such family. It contains $Z=A\cup (Z\setminus A)$ for any $A\in \mathcal P$. If $B\subset A$ belong to $\mathcal L$, so does $A\setminus B$, the complement of the disjoint union of $Z\setminus A$ and $B$. For $A\in \mathcal P$, the sets $B\in \mathcal L$ with $A\cap B\in \mathcal L$ form a family which contains $\mathcal P$ and is closed under countable disjoint unions and under complements, because $A\cap (Z\setminus B)=A\setminus (A\cap B)$; hence it equals $\mathcal L$. Thus $A\cap B\in \mathcal L$ whenever $A\in \mathcal P$ and $B\in \mathcal L$. Repeating the argument with $A\in \mathcal L$ shows that $\mathcal L$ is closed under finite intersections. Writing a countable union $\bigcup _ jA_ j$ of members of $\mathcal L$ as the disjoint union of the sets $A_ j\cap (Z\setminus A_1)\cap \cdots \cap (Z\setminus A_{j-1})$, we see that $\mathcal L$ is a $\sigma $-algebra. Hence $\sigma (\mathcal P)\subset \mathcal L$, and $\mathcal L$ is contained in every family as in (1).

(2) The sets $E\in \sigma (\mathcal P)$ with $\lambda (E)=\lambda '(E)$ form a family which contains $\mathcal P$, is closed under countable disjoint unions by countable additivity, and is closed under complements because $\lambda (Z\setminus E)=\lambda (Z)-\lambda (E)$ and $\lambda '(Z\setminus E)=\lambda '(Z)-\lambda '(E)$ are differences of finite numbers. By (1), this family is all of $\sigma (\mathcal P)$. $\square$