Let $(X,\mu)$ be a probability space, and $0<\epsilon<1/2$. Let $\{A\_i:i\in \mathbb{N}\}$ be a collection of measurable subsets of $X$ such that $\mu(A\_i)\geq \epsilon$ for all $i\in\mathbb{N}$.


Is it always true that there are indices $i<j$ such that $\mu(A\_i\cap A\_j)\geq \epsilon^2$ ? Is it possible to classify the (possible) counterexamples?



Measure of intersections in probability spaces
Take Borel measure on $[0,1]$ as an example. Cut off disjoint intervals $I\_1,I\_2,\dots$ where $I\_i$ has length $2^{-i}\epsilon$. That's length $\epsilon$ altogether. In the remaining $1-\epsilon$, take independent events $B\_1,B\_2,\ldots$ with $B\_i$ of measure $(1-2^{-i})\epsilon$. Define $A\_i=I\_i\cup B\_i$. Then $A\_i$ has measure $\epsilon$ for all $i$, and $A\_i\cap A\_j$ has measure $(1-2^{-i})(1-2^{-j})\epsilon^2\lt \epsilon^2$ for $i\ne j$.


Do I need to cite a theorem that $B\_1,B\_2,\ldots$ exist? I think it is standard elementary probability, and anyway it is easy to prove using finite unions of intervals.


