Let $X$ be a set. A \reftext{def:family-subfamily-subsets-set-2026a}{family} $\mathcal{P}$ of subsets of $X$ is a \textbf{$\pi$-system} if it is nonempty and closed under finite intersections: $A,B\in\mathcal{P}$ implies $A\cap B\in\mathcal{P}$.

A family $\mathcal{L}$ of subsets of $X$ is a \textbf{$\lambda$-system} if: (1) $X\in\mathcal{L}$; (2) if $A,B\in\mathcal{L}$ and $A\subseteq B$, then the \reftext{def:complement-subset-relative-set-2026a}{relative complement} $B\setminus A\in\mathcal{L}$; (3) for every nondecreasing \reftext{def:sequence-in-set-2026a}{sequence} $(A_m)_{m\in\mathbb{N}}$ in $\mathcal{L}$ (that is, $A_m\subseteq A_{m+1}$ for all $m$), $\bigcup_m A_m\in\mathcal{L}$.

\textbf{Theorem.} If $\mathcal{P}$ is a $\pi$-system, $\mathcal{L}$ is a $\lambda$-system, and $\mathcal{P}\subseteq\mathcal{L}$, then the \reftext{def:generated-sigma-algebra-2026a}{generated $\sigma$-algebra} satisfies