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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.)

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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. (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, you may want to take a look. (And I see Andreas just posted an answer while I was writing this.)