Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 1 / 25J / b

Every sigma-algebra is both a pi-system and a Dynkin system. Conversely, in a family having both properties, intersections and complements give finite unions and differences. For any sequence $(A_n)$ in it, replace $A_n$ by the disjoint sets $B_n=A_n\setminus {\bigcup }_{j<n}{A}_j$, which still belong to the family. Their disjoint union equals ${\bigcup }_n{A}_n$, proving closure under arbitrary countable unions. Thus the family is a sigma-algebra.