# A sequence of singular measures converging weakly* to a continuous measure

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?

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## 2 Answers

$$\mu_n = \sum_{i=1}^n \frac{1}{n} \delta_{\frac{i}{n}}$$

• Dear Nate, Pleasingly succinct! Best wishes, – Matt E Mar 10 '12 at 4:54
• +1. What else. – Did Mar 10 '12 at 10:07

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 $dx$.

• A reference for this equidistributed sequences besides wikipedia? – checkmath Mar 10 '12 at 4:59