From very many sets of fixed measure in a probability space, can we select many that have a positive intersection?

Question

I assume the following Lemma is either well known or, more probably, a Corollary of a much stronger well known Theorem, and I would be grateful for a reference:

For all $\delta\in (0,1)$ and all $\ell\in \mathbb N$ there are $M$ and $\varepsilon>0 $ such that: Whenever we have a probability space $\Omega$ and a family $(A_i:i\lt M)$ of sets of measure $\ge \delta$, we can find a subfamily of $\ell$ many sets whose intersection has measure at least $\varepsilon$.

PS: An (easy) proof is a nice but straightforward application of Ramsey's theorem.

PS2: Only finite additivity of the measure is required/relevant.

Answer 1

As Bill Johnson pointed out in his comment, the Lemma that I stated is trivial: simple counting is enough. Therefore, a reference is not required and would not make sense. So Bill Johnson answered my question (thanks!).

I would like to add a remark (unfortunately, a comment does not provide enough space, so I have to abuse an "answer" entry):

Originally we though of the following, slightly stronger variant of the Lemma (for which we used Ramsey's Theorem, but which might have a counting proof as well, who knows):

For all $\delta\in (0,1)$ and all $k\in \mathbb N$ there is an $\varepsilon(\delta,k)>0 $, and for all $\ell$ (bigger than $5/\delta$, say) there is a $M(\ell)$ such that: Whenever we have a probability space $\Omega$ and a family $\mathcal A=(A_i:i\lt N)$ of sets of measure $\ge \delta$, then we can find a subfamily $\mathcal B$ of size $\ell$ such that every subfamily $\mathcal C$ of $\mathcal B$ of size $k$ has an intersection of measure at least $\varepsilon$.

(So here $\varepsilon$ does not depend on $\ell$, but on $\delta$ and $k$ only. For $k=\ell$ we get the Lemma of the original question.)

Answer 2

This can be considered one version of the Poincare recurrence theorem. Vitaly Bergelson's paper "The multifarious Poincare recurrence theorem" might discuss something along these lines. See also the proof of Theorem 3.1 in Bergelson-Downarowicz's paper "Large sets of integers and hierarchy of mixing properties of measure-preserving systems" and p. 49 of Bergelson's "Ergodic Ramsey Theory - an Update" where a similar argument is used to prove Khintchine's recurrence Theorem.