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Here, the user Ramiro proves the Monotone Class Lemma from Dynkin's $\pi$-$\lambda$ theorem, and vice versa. However, in proving "monotone implies Dynkin" (there, "(2 $\Rightarrow$ 1)"), he uses the following:

[...] any class containing $\mathcal{C}$ which is closed under increasing limits and by difference contains $A(\mathcal{C})$ the algebra generated by $\mathcal{C}$.

($\mathcal C$ is a class of subsets of a universal set $\Omega$.)

I suspect that this is false: Otherwise, any $\lambda$-system, thereby containing the algebra generated by it, would contain finite unions and thus would be a $\sigma$-algebra. This is false, as shown in this post. Anyone can confirm this?

Despite the above statement, what Ramiro actually uses is that $$A(\mathcal C)\subseteq L(\mathcal C),$$ where $L(\mathcal C)$ is the $\lambda$-system generated by $\mathcal C)$. I am having trouble proving this. Can someone help?

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1 Answers1

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This is false in general. Consider the Dynkin $\lambda$-system $\mathcal{L}$ presented in the accepted answer here. If it were true that $\mathcal{A}(\mathcal{L})\subseteq \mathcal{L}$ then $\mathcal{L}$ would be closed under finite unions, but this is not true, a contradiction.

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