69 lines
3.1 KiB
TeX
69 lines
3.1 KiB
TeX
\section{Lambda Systems}
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\label{section:lambda-system}
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\begin{definition}[$\pi$-System]
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\label{definition:pi-system}
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Let $X$ be a set and $\mathcal{P} \subset 2^X$, then $\mathcal{P}$ is a \textbf{$\pi$-system} if:
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\begin{enumerate}
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\item[(P1)] $\emptyset \in \mathcal{P}$.
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\item[(P2)] For any $A, B \in \mathcal{P}$, $A \cap B \in \mathcal{P}$.
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\end{enumerate}
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\end{definition}
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\begin{definition}[$\lambda$-System]
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\label{definition:lambda-system}
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Let $X$ be a set and $\alg \subset 2^X$, then $\alg$ is a \textbf{$\lambda$-system/$d$-system} if:
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\begin{enumerate}
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\item[(L1)] $\emptyset, X \in \alg$.
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\item[(L2)] For any $E, F \in \alg$ with $E \subset F$, $F \setminus E \in \alg$.
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\item[(L3)] For any $\seq{A_n} \subset \alg$ with $A_n \subset A_{n+1}$ for all $n \in \nat^+$, $\bigcup_{n \in \nat^+}A_n \in \alg$.
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\end{enumerate}
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\end{definition}
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\begin{definition}[Generated $\lambda$-System]
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\label{definition:generated-lambda-system}
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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 $\ce$}.
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\end{definition}
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\begin{lemma}
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\label{lemma:pi-lambda}
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Let $X$ be a set and $\alg \subset 2^X$, then the following are equivalent:
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\begin{enumerate}
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\item $\alg$ is a $\sigma$-algebra.
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\item $\alg$ is a $\pi$-system and a $\lambda$-system.
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\end{enumerate}
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\end{lemma}
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\begin{proof}
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$(2) \Rightarrow (1)$: 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.
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\end{proof}
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\begin{theorem}[Dynkin's $\pi$-$\lambda$ Theorem]
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\label{theorem:pi-lambda}
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Let $X$ be a set and $\mathcal{P} \subset 2^X$ be a $\pi$-system, then $\sigma(\mathcal{P}) = \lambda(\mathcal{P})$.
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\end{theorem}
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\begin{proof}
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Let $\ce \subset \lambda(\mathcal{P})$ and
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\[
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\cm(\ce) = \bracs{E \in \lambda(\mathcal{P})| E \cap F \in \lambda(\mathcal{P}) \forall F \in \ce}
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\]
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then
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\begin{enumerate}
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\item[(L2)] Let $E, F \in \cm$ with $E \subset F$, then for any $G \in \ce$,
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\[
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(F \setminus E) \cap G = (F \cap G) \setminus (E \cap G) \in \lambda(\mathcal{P})
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\]
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\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
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\[
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\braks{\bigcup_{n \in \natp}E_n} \cap F = \bigcup_{n \in \natp}E_n \cap F \in \lambda(\mathcal{P})
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\]
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\end\{enumerate\}
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so $\cm(\ce)$ is a $\lambda$-system.
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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.
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\end{proof}
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