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From Rudin, Real and Complex Analysis, Chapter 8, Problem 7, 1st Edition.

Suppose $E$ is a compact set in $\mathbb{R}^{k}$ without isolated points. Show that $E$ is the support of a continuous positive Borel measure $\mu$. If $m(E)=0$, this gives examples of singular measures.

I am wondering how to prove this. For a classical Cantor set or even fat Cantor set, I know how to construct the measure; for similar fractal shapes analgous methods are possible. However if we only know $E$ is closed and bounded, every point is a limit point of $E$, then I am at a loss how to prove this fact.

Dimension is a barrier. Locally $E$ can be both "continuous" like closed cubes, or "discrete" like fat cantor sets. In the one dimensional case, constructing a sequence of continuous Borel measures via characteristic functions and taking the limit is usually suffice; in the general $\mathbb{R}^{k}$ case, I do not know what canonical choices I have, even though measure - function relationship still preserves.

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You can take a dense subset of $E$ to construct a sequence of measures that converges to a measure with support $E$ in the weak*-topology. –  Michael Greinecker Jan 11 '13 at 10:18
    
But how I know the measure I get the "dense subset of $E$" is continuous in the first place? –  Bombyx mori Jan 11 '13 at 10:20
    
I think the following works: Pick an enumeration of $E$ and let $\mu_n$ be the uniform distribution over the first $n$ points in the enumeration. –  Michael Greinecker Jan 11 '13 at 10:22
    
This should work only if $E$ is countable. –  Bombyx mori Jan 11 '13 at 10:24
    
Sorry, I meant an enumeration of a dense subset. –  Michael Greinecker Jan 11 '13 at 10:27
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