Let $X$ be a set. If $\mathcal{A}$ is a family of subsets of $X$ then let $\sigma(\mathcal{A})$ denote the $\sigma$-algebra generated by $\mathcal{A}$.

Definition $1$

A $\pi$-system on $X$ to be a non-empty family $\mathcal{A}\subseteq \mathcal{P}(X)$ closed under finite intersections (I wonder why such a fancy name).

Definition $2$

A $\lambda$-system on $X$ is a family $\mathcal{A}\subseteq \mathcal{P}(X)$ such that $X\in\mathcal{A}$ and that is closed under complements and unions of pairwise disjoint countable subfamilies.

Definition $3$

A monotone class on $X$ is a family $\mathcal{A}\subseteq \mathcal{P}(X)$ that is closed unders unions and intersections of countable subfamilies of nested sets.

Dynkin's $\pi-\lambda$ Theorem (source)

Let $\mathcal{I}$ be a $\pi$-system on $X$ and $\mathcal{D}$ be a $\lambda$-system on $X$. If $\mathcal{I}\subseteq \mathcal{D}$ then $\sigma(\mathcal{I})\subseteq \mathcal{D}$.


The Monotone Class Theorem (source)

Let $\mathcal{A}$ be an algebra of subsets of $X$ and suppose that $\mathcal{M}$ is the smallest monotone class on $X$ such that $\mathcal{A}\subseteq \mathcal{M}$, then $\sigma(\mathcal{A})=\mathcal{M}$.

My question:

I'd like to know if these two theorems are equivalent, in the sense of whether one can prove easily one from the other, in a way that makes it clear that for practical purposes they are to some extent interchangeable. More formally I would raise the question of whether they are equivalent given some "basic" set theory axioms (e.g. ZF).

This question seems to be asking the same thing but it's actually considering a different statement for the Monotone Class Theorem.



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