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](https://arxiv.org/abs/1605.08580), 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.



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


