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. 



Intersection of an uncountable number of sets.
If the uncountable number of sets that you're intersecting is small enough, you *might* be able to guarantee that the intersection is measurable (and in fact of measure 1) --- it depends on information about the set-theoretic universe that the usual axioms (ZFC) don't decide. Specifically, the *additivity of measure* is defined to be the smallest cardinal $\kappa$ such that some $\kappa$ sets of Lebesgue measure 0 have a union that is not of Lebesgue measure 0. Obviously, this cardinal is at least $\aleph\_1$ and at most the cardinality $\mathfrak c$ of the continuum. So if the continuum hypothesis holds, there's nothing more to be said about it. But in the absence of the continuum hypothesis, the additivity of measure is as small as $\aleph\_1$ in some models of set theory, as large as $\mathfrak c$ in other models, and in between in still other models. If you intersect strictly fewer than this many sets of measure 1, the intersection will have measure 1. Whether that fact covers any uncountable famlilies of sets is, as indicated above, not decided by ZFC. 


