\section{Lambda Systems}
\label{section:lambda-system}

\begin{definition}[$\pi$-System]
\label{definition:pi-system}
    Let $X$ be a set and $\mathcal{P} \subset 2^X$, then $\mathcal{P}$ is a \textbf{$\pi$-system} if:
    \begin{enumerate}
        \item[(P1)] $\emptyset \in \mathcal{P}$.
        \item[(P2)] For any $A, B \in \mathcal{P}$, $A \cap B \in \mathcal{P}$.
    \end{enumerate}
\end{definition}

\begin{definition}[$\lambda$-System]
\label{definition:lambda-system}
    Let $X$ be a set and $\alg \subset 2^X$, then $\alg$ is a \textbf{$\lambda$-system/$d$-system} if:
    \begin{enumerate}
        \item[(L1)] $\emptyset, X \in \alg$.
        \item[(L2)] For any $E, F \in \alg$ with $E \subset F$, $F \setminus E \in \alg$.
        \item[(L3)] For any $\seq{A_n} \subset \alg$ with $A_n \subset A_{n+1}$ for all $n \in \nat^+$, $\bigcup_{n \in \nat^+}A_n \in \alg$.
    \end{enumerate}
\end{definition}

\begin{definition}[Generated $\lambda$-System]
\label{definition:generated-lambda-system}
    Let $X$ be a set and $\ce \subset 2^X$, then the smallest $\lambda$-system $\lambda(\ce)$ over $X$ containing $\ce$ is the \textbf{$\lambda$-system generated by $\ce$}.
\end{definition}

\begin{lemma}
\label{lemma:pi-lambda}
    Let $X$ be a set and $\alg \subset 2^X$, then the following are equivalent:
    \begin{enumerate}
        \item $\alg$ is a $\sigma$-algebra.
        \item $\alg$ is a $\pi$-system and a $\lambda$-system.
    \end{enumerate}
\end{lemma}
\begin{proof}
    $(2) \Rightarrow (1)$: Let $A, B \in \alg$, then $A \cup B = (A^c \cap B^c)^c \in \alg$. Thus $\alg$ is an algebra. By \autoref{lemma:sigma-algebra-condition}, $\alg$ is a $\sigma$-algebra.
\end{proof}


\begin{theorem}[Dynkin's $\pi$-$\lambda$ Theorem]
\label{theorem:pi-lambda}
    Let $X$ be a set and $\mathcal{P} \subset 2^X$ be a $\pi$-system, then $\sigma(\mathcal{P}) = \lambda(\mathcal{P})$.
\end{theorem}
\begin{proof}
    Let $\ce \subset \lambda(\mathcal{P})$ and
    \[
    \cm(\ce) = \bracs{E \in \lambda(\mathcal{P})| E \cap F \in \lambda(\mathcal{P}) \forall F \in \ce}
    \]

    then
    \begin{enumerate}
        \item[(L2)] Let $E, F \in \cm$ with $E \subset F$, then for any $G \in \ce$,
        \[
        (F \setminus E) \cap G = (F \cap G) \setminus (E \cap G) \in \lambda(\mathcal{P})
        \]

        \item[(L3)] Let $\seq{E_n} \in \cm$ with $E_n \subset E_{n+1}$ for all $n \in \natp$, and $F \in \ce$, then
        \[
        \braks{\bigcup_{n \in \natp}E_n} \cap F = \bigcup_{n \in \natp}E_n \cap F \in \lambda(\mathcal{P})
        \]

    \end\{enumerate\}

    so $\cm(\ce)$ is a $\lambda$-system.

    Since $\mathcal{P}$ is a $\pi$-system, $\cm(\mathcal{P}) \supset \mathcal{P}$, so $\cm(\mathcal{P}) = \lambda(\mathcal{P})$. Thus for any $E \in \lambda(\mathcal{P})$ and $F \in \lambda(\mathcal{P})$, $E \cap F \in \lambda(\mathcal{P})$. Therefore $\cm(\lambda(\mathcal{P})) \supset \mathcal{P}$, $\cm(\lambda(\mathcal{P})) = \lambda(\mathcal{P})$, and $\lambda(\mathcal{P})$ satisfies (P2). By \autoref{lemma:pi-lambda}, $\lambda(\mathcal{P})$ is a $\sigma$-algebra.
\end{proof}
