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

Question

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?

Answer 1

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 Lebesgue measure $dx$.

Answer 2

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