# Monotone class theorem vs Dynkin $\pi-\lambda$ theorem

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?

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.

• Good. I was confused because my book calls the first "Monotone class theorem". Also on the wiki page the proof was the same of the one reported on my book (and it was therefore proving, instead of the monotone class theorem, the dynkin $\pi-\lambda$ theorem). I edited the proof out of the wiki page in the meanwhile :-) Thank you!
– Ant
Mar 17, 2015 at 15:11
• By the way, in the (second version of the) Dynkin $\pi$-$\lambda$ theorem above, we cannot conclude that $D$ is a $\sigma$-algebra. A counterexample is given by taking $P$ to be the trivial $\sigma$-algebra $\{\emptyset,\Omega\}$, and $D$ to be any $\lambda$-system that is not a $\sigma$-algebra (e.g., take $\Omega=\{1,2,3,4\}$ and $D=\{\emptyset,\{1,2\},\{3,4\}, \{1,3\}, \{2,4\}, \Omega\}$. Mar 17, 2015 at 15:17
• Ah, I see. Thanks for pointing that out! :)
– Ant
Mar 17, 2015 at 15:18
• Kallenberg's Foundations of Modern Probability is another book that uses the term "monotone class argument" to refer to the Dynkin $\pi$-$\lambda$ theorem, so apparently this isn't such uncommon usage, but I agree that it is confusing and should probably be avoided. Mar 17, 2015 at 15:22
• I'd be interested in knowing whether the $\pi - \lambda$ Theorem and the Monotone class Theorem are also equivalent. Jul 9, 2017 at 10:16