# Conditions for non-triviality of Caratheodory measure

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.

• Of course $S$ should be infinite. – Sungjin Kim May 3 '12 at 1:02
• @i707107: why do you think it is necessary? – Ilya May 3 '12 at 11:21
• I meant that if you want the sigma algebra to be infinite. – Sungjin Kim May 4 '12 at 5:28
• I would like to know the answer, too. – Mahdi Majidi-Zolbanin May 19 '12 at 3:03
• Another question on "non-triviality" of a Carathéodory measure: is there a set with measure other than $0$ or $\infty$? It is a non-trivial result for Hausdorff measures that (under the right conditions) a set with infinite measure has a subset with positive finite measure. – Gerald Edgar Jun 22 '12 at 13:44

• Very interesting. For a Galois connection we require partial orders on the collection of positive measures and the collection of outer measures. Do you have an idea which partial order to choose, so that the fixed points of the Galois connection are precisely the "Caratheodory complete" measures (i.e. complete and defined on the $\sigma$-algebra of Caratheodory measurable sets) and the regular outer measures? – yada Oct 15 '19 at 12:54
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.