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

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.

-
Welcome to mathoverflow! What a coincidence, I was just thinking about this very question :-) –  Goldstern Sep 28 '12 at 22:13
Welcome, Jakob. –  Andres Caicedo Sep 28 '12 at 22:28

-