3
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ What about Dirac measures? $\endgroup$ – Gabe Conant Aug 13 at 14:19
  • $\begingroup$ Oh I see, you are asking for the support of the measure to be $X$ $\endgroup$ – Gabe Conant Aug 13 at 14:23
  • 6
    $\begingroup$ $\sum \lambda_n \delta_{t_n}$ for a positive $\ell^1$ sequence of scalars and a dense sequence? $\endgroup$ – user131781 Aug 13 at 14:38
  • 1
    $\begingroup$ Incidentally, a locally compact, second countable Hausdorff space is Polish. $\endgroup$ – Nate Eldredge Aug 13 at 16:19
  • 2
    $\begingroup$ 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 at 20:00
7
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.