The question is asked in a generality that makes it difficult to say any positive results. Part of the problem is that, even for Lebesgue measure on the interval ${}[0,1]$, the usual axioms of set theory do not suffice to give a full answer.
For example, if CH holds, then even if we restrict our attention to intersections indexed by $\omega_1$, the smallest uncountable cardinal, any set of reals can appear this way. In fact, we only need to consider sets $E_i$ of the form ${}[0,1]$, or ${}[0,1]\setminus\{x\}$. Even if CH fails, the same idea shows that we do not really want the intersections to have "too many indices", i.e., we want $|{\mathcal I}|<|{\mathbb R}|$.
Now, it is a consequence of the very useful Martin's axiom that every ${\mathcal I}$-intersection of measurable subsets of ${}[0,1]$ is measurable, as long as $|{\mathcal I}|<|{\mathbb R}|$. (On the other hand, it is consistent that CH fails and yet there are intersections of size $\omega_1$ that are not measurable, so one really needs to appeal to additional axioms to get such a general positive statement.)
The appropriate context for this problem seems to be the theory of cardinal invariants of the continuum, see Joel's very nice answer here; the relevant invariant being the additivity of Lebesgue measure. (That Martin's axiom gives the result above is because all (classical) cardinal invariants have size $|{\mathbb R}|$ under Martin's axiom.) Andreas Blass and Tomek Bartoszynski wrote excellent introductions to this topic for the Handbook of Set Theory and elsewhere, you may want to take a look at their papers. (And I see Andreas just posted an answer while I was writing this.)
