Past exam of the real and harmonic analysis course of the University of Oxford / 2007 / 2 / c

A metric outer measure on a metric space $(X,d)$ is an outer measure satisfyingHere $d(A,B)=\inf \{d(a,b):a\in A,b\in B\}$. Every Borel set is measurable for a metric outer measure. Consequently its measurable sigma-algebra contains the Borel sigma-algebra; it may be strictly larger.