given an outer measure $\eta: 2^X \to [0, \infty]$ we call a subset $E$ of $X$ $\eta$-measurable (measureable with respect to the outer measure), if for every subset $Q$ of $X$, the following holds:
$\eta(Q) = \eta(Q \cap E) + \eta(Q \cap E^C)$
so heres my question: do you know any examples of sets E that are not $\eta$-measurable? Thanks!