TITLE: From very many sets of fixed measure in a probability space, can we select many that have a positive intersection?
QUESTION [9 upvotes]: 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.

REPLY [3 votes]: 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.)