Dynkin's Pi-Lambda Theorem

Definition

$l(\mathcal{P})=\sigma (\mathcal{P})$ where $\mathcal{P}$ is a Pi-System

If $\mathcal{P}$ is a Pi-System, then the small Lambda-System containing $\mathcal{P}$, $l(\mathcal{P})$ is the sigma-field generated by $\mathcal{P}$, $\sigma (\mathcal{P})$.

$\sigma (\mathcal{P})\subset \mathcal{L}$ where $\mathcal{P}$ is a Pi-System and $\mathcal{L}$ is a Lambda-System containing $\mathcal{P}$

If $\mathcal{P}$ is a Pi-System and $\mathcal{L}$ is the Lambda-System containing $\mathcal{P}$, then $\sigma (\mathcal{P})\subset \mathcal{L}$

Proof

If $\mathcal{A}$ is a both Lambda-System and Pi-System, then $\mathcal{A}$ is a Sigma-Field

Since the condition of $\sigma (\mathcal{P})$ is more complex, then $l(\mathcal{P})$, $l(\mathcal{P})\subset \sigma (\mathcal{P})$ is satisfied

The $l(\mathcal{P})$ containing a Pi-System $\mathcal{P}$ is a Pi-System itself. If Lambda-System and Pi-System, then Sigma-Field Therefore Lambda-System $l(\mathcal{P})$ is a Sigma-Field

Since $\sigma (\mathcal{P})$ is the smallest Sigma-Field, $l(\mathcal{P})\supset \sigma (\mathcal{P})$ is satisfied

Summary

	$A\cap B$	$\varnothing$	$A-B$	${\cup }_i{A}_i=\underset{i}{⨄}{B}_i$	$\Omega$	$A^c$	$A\cup B$	$\bigcup _{i=1}^{\infty }{A}_i$	$\underset{i=1}{\overset{\infty }{⨄}}{B}_i$	$\bigcap _{i=1}^{\infty }{A}_i$
$\pi$-system	$O$	$X$	$X$	$X$	$X$	$X$	$X$	$X$	$X$	$X$
$\lambda$-system	$X$	$O$	${\Delta }^{\prime }$	$X$	$O$	$O$	$X$	$X$	$O$	$X$
$\sigma$-field	$O$	$O$	$O$	$O$	$O$	$O$	$O$	$O$	$O$	$O$