TITLE: Conditions for non-triviality of Caratheodory measure
QUESTION [8 upvotes]: This may be too vague to end up being useful, but: 
Are there any (natural? reasonable?) conditions that can be imposed on an outer measure $\phi^*:{\mathcal P}(S)\to[0,\infty]$ to ensure that the $\sigma$-algebra of measurable sets (obtained through Caratheodory's construction) is non-trivial? 
By non-trivial, I mean that at least it contains some sets other than $\emptyset,S$; even better if the condition ensures the $\sigma$-algebra is infinite.

REPLY [6 votes]: One commonly used condition, even found in the original paper of Caratheodory, is what we now call a metric outer measure.  $(S,d)$ is a metric space.  A metric outer measure is an outer measure $\mu$ such that: If two sets $A, B \subseteq S$ have positive distance (there is $\delta > 0$ so that for all $a \in A$ and $b \in  B$ we have $d(a,b) \ge \delta$), then $\mu(A \cup B) = \mu(A) + \mu(B)$.  Then: if $\mu$ is a metric outer measure, then at least all Borel sets in $S$ are $\mu$-measurable sets.