# Does there exist a non-zero signed finite borel measure which is zero on all balls?

Let $$(X,d)$$ be a compact separable metric space. Let $$\mu$$ be a Borel, regular, finite, signed measure on $$X$$ such that for all $$x\in X$$, for all $$r>0$$, $$\mu(B(x,r))=0$$, where $$B$$ denotes the (either open or closed) ball w.r.t $$d$$.

Is $$\mu$$ zero?

If $$\mu$$ is positive one can show that $$\mu=0$$ using the Borel-Lebesgue theorem, but what if $$\mu$$ is signed?

• The set of balls generates the Borel sigma-algebra, so that $\mu$ is characterized by its values on balls and must be zero (if you are more comfortable with positive measures, think about the decomposition of $\mu$ into its positive and negative part, and observe that they have the same values on all balls and thus coincide). – Benoît Kloeckner Mar 4 at 17:17
• @BenoîtKloeckner Erm... How do you show that the family of zero measure sets is a $\sigma$-algebra? Usually you can do it if you start with a semi-ring using the monotone class lemma, but balls do not form one. Am I missing anything? – fedja Mar 4 at 17:44
• @BenoîtKloeckner: The usual argument for this needs a collection which not only generates the Borel $\sigma$-algebra but is also closed under finite intersections (a $\pi$-system), which the balls do not satisfy. – Nate Eldredge Mar 4 at 18:19
• @fedja and NateElredge I have been naïve, thanks for the correction and sorry for my erroneous comment.. – Benoît Kloeckner Mar 5 at 20:02

the author constructs a compact metric space $$X$$ and two distinct Borel probability measures $$\mu_1, \mu_2$$ that agree on every closed ball. (They must therefore also agree on open balls, because an open ball is a countable increasing union of closed balls.) Taking $$\mu = \mu_1 - \mu_2$$ provides your desired signed measure.
• On the positive side, the answer to whether $\mu$ is zero is yes for Banach spaces; as proved by D. Preiss and J. Tiser, Measures in Banach spaces are determined by their values on balls. Mathematika 38, No. 2, 391-397 (1991). Zbl 0755.28006 – Dirk Werner Mar 7 at 17:33