Can anyone provide a sequence of singular (w.r.t. Lebesgue measure) measures $\in\mathcal{M}([0,1])=C[0,1]^*$ converging $weakly^*$ to an absolutely continuous (w.r.t. Lebesgue measure) measure?
A sequence of singular measures converging weakly* to a continuous measure
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real-analysis
functional-analysis
measure-theory
singular-measures
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1Two small requests: 1. If you re-post a question, either [here](http://math.stackexchange.com/questions/118061/radon-nikodym-decomposition) or [on MO](http://mathoverflow.net/questions/90654/about-a-radon-measure-closed) please say that you asked the same (or a very closely related one) elsewhere. 2. Please don't introduce new tags, stick to those that are available. – 2012-03-10
2 Answers
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$\mu_n = \sum_{i=1}^n \frac{1}{n} \delta_{\frac{i}{n}}$
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0+1. [What else](http://www.youtube.com/watch?v=2Ts04dO2RKQ). – 2012-03-10
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If $(x_n)$ is a sequence of points in $[0,1]$, then $x_n$ is equidistributed with respect to the absolutely continuous measure $f(x)dx$ (where $dx$ denotes Lebesgue measure) precisely if the sequence of singular measures $\dfrac{1}{n} \sum_n \delta(x-x_n)$ on $[0,1]$ converges in the weak-$*$ topology to the measure $f(x)dx$.
For example, if $(x_n)$ is the sequence $(\alpha n \bmod 1)$ for an irrational number $\alpha$ then $(x_n)$ is equidistributed with respect to Lebesque measure $dx$.
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0A reference for this equidistributed sequences besides wikipedia? – 2012-03-10