There is a standard definition for measure theory which defines measurable sets for an outer measure $\mu^*$ as those which have the property that:
$\mu(E) = \mu(E\cap A) + \mu(E \cap A^c)$
My question is, why should we require the measurable sets behave well with all sets instead of those in a strictly smaller subset?
For example, why not restrict $E$ to be an element of the borel sigma algebra instead of a general subset of $R$?