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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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    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 $1/2$.2012-03-25

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