Let $\mathcal{I}$ be an uncountable set. Let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space, and $E_i, i\in \mathcal{I}$ be a measurable set such that $\mathbb{P}(E_i)=1$. What can we say about $\mathbb{P}(\cap_{i\in \mathcal{I}} E_i)?$.

I know that an uncountable intersection of measurable sets is not necessarily measurable. But are there in general conditions that allow such infinite intersections to be measurable?

Apologies if the question is too elementary or unclear.