# Must a locally compact, second countable, Hausdorff space support a Radon measure?

Let $$X$$ be a locally compact, second countable and Hausdorff space, must there be a Radon measure on $$X$$ whose support is $$X$$?

The motivation for this question comes from Anton Deitmar's paper On Haar systems for groupoids, in which he construct a groupoid with open range map admitting no Haar system starting from a locally compact (actually compact) Hausdorff space that supports no Radon measures. Deitmar then conjectures that all locally compact, second countable and Hausdorff groupoids with an open range map have a Haar system, so I believe that either there are no known counterexamples or that the answer to my question is positive, but I haven't been able to find neither a counterexample nor a proof.

• What about Dirac measures? – Gabe Conant Aug 13 '19 at 14:19
• Oh I see, you are asking for the support of the measure to be $X$ – Gabe Conant Aug 13 '19 at 14:23
• $\sum \lambda_n \delta_{t_n}$ for a positive $\ell^1$ sequence of scalars and a dense sequence? – user131781 Aug 13 '19 at 14:38
• Incidentally, a locally compact, second countable Hausdorff space is Polish. – Nate Eldredge Aug 13 '19 at 16:19
• Also, if $X$ has no isolated points, you can even find a full-support measure which is atomless. Let $U_n$ be a countable base. Since $X$ is Polish, each $U_n$ contains a copy of Cantor space $2^\omega$, on which you can put Cantor (aka Lebesgue) measure, $\mu_n$. Now consider $\mu = \sum 2^{-n} \mu_n$. – Nate Eldredge Aug 13 '19 at 20:00

At the OP‘s request——consider $$\sum \lambda_n \delta_{t_n}$$ where $$(\lambda_n)$$ is a sequence of positive scalars which sum to $$1$$ and $$(t_n)$$ is a dense sequence.