When introducing measurability, we noted that we wanted the following property to hold for disjoint $A, B \in \mathcal{P}(\mathbb{R})$
$m(A \cup B) = m(A)+m(B)$ (additivity)
We then defined a set A to be measurable iff $\forall$ sets $E \in \mathcal{P}(\mathbb{R}),$
$m(E) = m(E \cap A)+m(E \cap A^c)$
This begs the question: Are the measurable sets the (unique) maximal collection of sets from $\mathcal{P}(\mathbb{R})$ which are additive? If so, our "goal" is certainly achieved. Of course it is also a question whether such a maximal set is even unique.