This problem is for my intro to real analysis class and I'm really struggling with some of the involved concepts with this problem.
If $E_1, E_2,...,E_n$ are disjoint measurable sets, how do you show that $$m^* (A\cap [\cup E_i]) = \sum m^*(A\cap E_i)$$
I think I understand the properties of sets and their measures that make this true but I have no clue how to go about proving it.
The key aspect of measurability with respect to an outer measure is that a measurable set $E$ 'splits' a set $A$ additively in the sense that if $E$ is measurable, we have $m^*A = m^*(A\cap E) + m^*(A \setminus E)$ for any set $A$.
Let $\Delta = A \cap \cup_k E_k = \cap_k (A \cap E_k)$, where the $E_k$ are pairwise disjoint. Note that $\Delta \cap E_k = A \cap E_k$.
Then $m^* \Delta = m^* (\Delta \cap E_1) + m^*(\Delta \setminus E_1)= m^* (A \cap E_1) + m^*(\Delta \setminus E_1)$.
Now note that $\Delta \setminus E_1 = A \cap \cup_{k\ge 2} E_k $, and repeat.
Hence $m^* (A \cap \cup_k E_k) = \sum_k m^* (A \cap E_k)$.