Dynkin Pi-Lambda Theorem

definition: a system of sets of a set $X$ is a collection of subsets of $X$ . A $\pi$ -system is a system of sets that is closed under intersection. A $\lambda$ -system is a system of sets that contains $X$ , is closed under monotone set difference and upper countable limits. A $\sigma$ -algebra is a system contaning $X$ , closed under complementation and countable union.

lemma: $H$ is a $\pi,\lambda$ -system iff $H$ is a $\sigma$ -algebra.

proof: Suppose $H$ is a $\pi,\lambda$ -system. Then it contains $X$ . Let $a$ be any set of $H$ . Then $a^C = X-a$ is a monotone set difference of sets of $H$ . Hence $H$ is closed under complement. Since $H$ is a $\pi$ -system that is closed under complement, it is closed under finite unions. Let $\left< A_i \right>$ be a sequence of sets of $H$ . Then the sequence $B$ defined by $B_i = \cup_{j = 1..i} A_j$ is a non-decreasing sequence of sets of $H$ and therefore contains the limit $\cup_i B_i = \cup_i A_i$ . Hence $H$ is a $\sigma$ -algebra.

Conversely, suppose $H$ is a $\sigma$ -algebra. Since it is closed under countable union and contains the $\emptyset = X^C$ , it follows that it is closed under finite union. Since it is also closed under complementation, it is closed under finite intersection and therefore is a $\pi$ -system. Let $a,b \in H$ be such that $a \subseteq b$ . Then $b - a = b \cap a^C \in H$ and so $H$ is closed under monotone set difference. Let $A$ be a non-decreasing sequence of sets of $H$ . Then the union $\cup_i A_i$ is an element of $H$ and so $H$ is a $\lambda$ -system.

The following theorem an an important abstract result used to prove a number of results in measure theory and is called Dynkin’s $\pi,\lambda$ theorem.

theorem: If $H$ is a $\lambda$ -system containing the $\pi$ -system $P$ , then $H$ contains $\lambda(P)$ .

proof: Let $G(a)$ be the system of sets such that $b \in G(a)$ iff $a \cap b \in \lambda(P)$ . When $a \in \lambda(P)$ then $G(a)$ is a $\lambda$ -system: $X \in G(a)$ since $X \cap a = a \in \lambda(P)$ , when $c,d \in G(a)$ such that $c \subseteq d$ , $(d - c) \cap a = (d \cap a) - (c \cap a)$ and $d \cap a, c \cap a \in \lambda(P)$ and $c \cap a \subseteq d \cap a$ it follows that $(d - a) \cap (c-a) \in \lambda(P)$ . Finally it $B$ is a non-decreasing sequence of sets of $G(a)$ then the sequence $C$ such that $C_i = B_i \cap a$ is a non-decreasing sequence of sets of $\lambda(P)$ and so $\cup_i C_i \in \lambda(P)$ . But $\cup_i C_i = (\cup_i B_i) \cap a$ and so $\cup_i B_i \in G(a)$ .

If $a \in P$ , then when $b \in P$ , $b \in G(a)$ since $P$ is a $\pi$ -system and $P$ is a subsystem of $\lambda(P)$ . But since $a \in \lambda(P)$ , $G(a)$ is a $\lambda$ -system that contains $P$ , it follows that $G(a)$ contains $\lambda(P)$ . But this implies when $a \in \lambda P$ , $G(a)$ contains $P$ and so $G(a)$ contains $\lambda P$ . Hence $\lambda(P)$ is closed under intersection and so is a $\pi$ -system.

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