TITLE: A set of positive measure contains a product set of positive measure?
QUESTION [10 upvotes]: The following question arose in my research on variations on Bell's theorem. I have tried to solve it on my own, but my weak background in measure theory apparently doesn't allow me to do so within a reasonable amount of time.
This is my first post on any SE site. Since the question is probably not research-level, I'm posting it here instead of on MO.

Let $(\Omega_1,\mathcal{F}_1,P_1)$ and $(\Omega_2,\mathcal{F}_2,P_2)$ be probability spaces. The product $\Omega_1\times\Omega_2$ comes equipped with the standard product $\sigma$-algebra and product measure.
If $A\subseteq \Omega_1\times\Omega_2$ is of positive measure, do there exist $B_1\subseteq\Omega_1$ and $B_2\subseteq\Omega_2$ of positive measure such that $B_1\times B_2\subseteq A$?

If this turns out to be false, then what about the same question with $B_1\times B_2\subseteq_{a.s.} A$ instead of exact containment?
Edit: I have accepted @leslie's answer as it resolves the original problem. I still hope for a positive answer to the revised question, where I allow $A$ to be modified by a set of measure zero. Can anyone say anything about this?

REPLY [9 votes]: One counterexample is the subset of $[0,1] \times [0,1]$ (with the usual Lebesgue $\sigma$-algebras on both copies of $[0,1]$) given by
$$
E = \{(x,y) \in [0,1] \times [0,1]: y - x \not \in \mathbb{Q}\}.
$$
It turns out that $E$ has planar measure $1$, but $E$ does not contain any cylinder set of the form $A \times B$ with $A,B$ Lebesgue measurable sets of positive measure.  One way to see this is to appeal to the nontrivial but better known fact that if $A$ and $B$ are Lebesgue measurable subsets of $\mathbb{R}$ with positive measure, the difference set $A - B = \{a - b: a \in A, b \in B\}$ must contain a nontrivial open interval (so, in particular, rational numbers).  The special case of this assertion when the sets $A$ and $B$ are the same is very well known and apparently originally due to Steinhaus.
I recalled this example from Falconer's The Geometry of Fractal Sets, where it is Exercise 5.4 (and the generalization of Steinhaus's observation is Exercise 1.7).