What is this property of Borel measures called? Consider a Borel measure (on a metric space) that is nonzero on all uncountable Borel subsets of the space, and zero on all countable Borel subsets of the space.
Is there a specific name for this property of the measure? I would like to give it a name in a text I'm writing, but that would of course be unnecessary if it already has one.
 A: We can show that the only measure satisfying this property on a Polish space (i.e. a separable topological space that admits a complete metric) is the trivial one (i.e. $0$ on countable sets and $\infty$ on uncountable ones).
Define a Borel isomoprhism to be a measurable bijection with a measurable inverse.
Theorem 8.3.6 of Donald L. Cohn's Measure Theory states that any two uncountable subsets of a Polish space are Borel isomorphic.
It is also a standard result that any two Polish spaces are Borel isomorphic. Since Borel isomorphic measurable spaces admit the same collection of measures, we can WLOG consider putting a Borel measure on $\mathbb{R}^2$ with the standard topology. So suppose we have a Borel measure satisfying your property on $\mathbb{R}^2$. Then since $\mathbb{R}^2$ is uncountable, every uncountable Borel set of $\mathbb{R}^2$ is Borel isomorphic to $\mathbb{R}^2$. Then note that $\mathbb{R}^2$ can be written as the uncountable disjoint union of uncountable Borel sets (consider translates of the $x$-axis), thus any uncountable Borel subset of $\mathbb{R}^2$ can be written as the uncountable disjoint union of uncountable Borel sets. Thus every uncountable Borel set has infinite measure. Thus the measure is the trivial one.
