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what is the meaning of every Borel probability measure on a compact Hausdorff space is the pointwise limit of a net of discrete probability measures, each having the same barycenter?

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Which one of the words causes trouble? –  t.b. Mar 25 '12 at 12:59
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To understand "barycenter" you need more structure than "compact Hausdorff space" perhaps. Maybe start with a simple case. A Borel probability measure on $[0,1]$ with mean $1/2$ is the limit (in the narrow topology for measures) of a sequence of discrete probability measures, each having mean $1/2$. –  GEdgar Mar 25 '12 at 13:26

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