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.