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.

`$|M| \delta$`

and sup norm at most`|M|$`

. – Bill Johnson Sep 28 '12 at 23:22