Consider the measure space $(\Omega,\mathcal F,P)$ where $\Omega=\{0,1\}^{\mathbb N}$ and $\mathcal F$ is the product Borel $\sigma$-algebra and $P$ is the law of an independent sequence of Bernoulli$(1/2)$ random variables. Let $$\mathcal A=\{A\in \mathcal F\colon P(A)=1\}$$ denote the collection of all almost sure events. Is the intersection of all sets in $\mathcal A$ non-empty?
I can't decide if it is empty or non-empty. On the one hand, it is easy to rule out many sequences from lying in the intersection: such a sequence cannot be eventually periodic, yet the set of ones in any such sequence must have density $1/2$ in $\mathbb N$. On the other hand, the (topological) support of the measure $P$ with respect to the product topology consists of all binary sequences (since every cylinder set containing a given sequence has positive measure), and one might expect that the support of $P$ is the closure of the intersection of $\mathcal A$.