Given an outer measure $\mu^{*}:(X,d) \rightarrow [0, \infty]$, where $X$ is a metric space, will all Borel Sets be $\mu^{*}$-measurable only when $\mu^{*}$ is a metric outer measure? Because if $\mu^{*}$ is just an ordinary outer measure then I don't see how this would follow?
(Obviously in practice, almost all outer measures we deal with like the Lebesgue outer measure, Hausdorff outer measure and so on are metric outer measures)
Best Answer
If $\mu^*$ is a metric outer measure, then all Borel Sets will be $\mu^*$-measurable, but if we have no restriction to $\mu^*$, then we can say nothing about Borel Sets. For instance, take $\mu^*$ the outer measure giving rise to the Lebesgue measure in $\mathbb{R}$, but take in $\mathbb{R}$ the discrete topology (which is a metric topology with distances 0 or 1). Then every sets is Borel, but not necesarily measurable.