= Solution
An <outer measure> on $X$ is a function $\mu:\mathcal P(X)\to[0,\infty]$ satisfying $\mu(\varnothing)=0$, <monotonicity>, and <countable subadditivity>:
$$
\mu\left(\bigcup_jE_j\right)\leq\sum_j\mu(E_j).
$$
Its <measurable sets> are those satisfying <Caratheodory measurability criterion>:
$$
\boxed{\mathfrak M_\mu^*=\{A\subseteq X:\mu(E)=\mu(E\cap A)+\mu(E\setminus A)\text{ for every }E\subseteq X\}.}
$$
The test runs over all subsets $E$, including nonmeasurable ones.
Back to article page
