Monotone class theorem:
Let $\mathcal C$ be a class of subset closed under finite
intersections and containing $\Omega$ (that is, $\mathcal C$ is a
$\pi$-system). Let $\mathcal B$ be the smallest class containing
$\mathcal C$ which is closed under increasing limits and by difference
(that is, $\mathcal B$ is the smallest $\lambda$ system containing $\mathcal C$). Then $\mathcal B =
\sigma(\mathcal C)$
Dynkin $\pi-\lambda$ theorem
If $P$ is a $\pi$ system and $D$ is a $\lambda$ system with $P
\subseteq D$, then $\sigma(P) \subseteq D$
(Also, I believe that it can concluded that $D$ is a $\sigma$ algebra)
It seems to me that they are basically the same thing. Dynking statement is slightly more general but more or less the same. Is it it true or am I misunderstanding something?
Best Answer
You're right that each of these theorems very quickly implies the other: You get the Dynkin $\pi$-$\lambda$ theorem by applying the monotone class theorem with $\mathcal C=\mathcal P$, and you get the converse by applying the Dynkin $\pi$-$\lambda$ theorem with $\mathcal P=\mathcal C$ and $D=\lambda(\mathcal C)$.
It might be preferable to call these both the Dynkin $\pi$-$\lambda$ theorem and reserve the name ``monotone class theorem" for the similar theorem which actually involves monotone classes.