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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.

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    $\begingroup$ What about Dirac measures? $\endgroup$
    – Gabe Conant
    Aug 13, 2019 at 14:19
  • $\begingroup$ Oh I see, you are asking for the support of the measure to be $X$ $\endgroup$
    – Gabe Conant
    Aug 13, 2019 at 14:23
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    $\begingroup$ $\sum \lambda_n \delta_{t_n}$ for a positive $\ell^1$ sequence of scalars and a dense sequence? $\endgroup$
    – user131781
    Aug 13, 2019 at 14:38
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    $\begingroup$ Incidentally, a locally compact, second countable Hausdorff space is Polish. $\endgroup$
    – Nate Eldredge
    Aug 13, 2019 at 16:19
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    $\begingroup$ 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 $\mu = \sum 2^{-n} \mu_n$. $\endgroup$
    – Nate Eldredge
    Aug 13, 2019 at 20:00

1 Answer 1

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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.

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