TITLE: A sequence of singular measures converging weakly* to a continuous measure
QUESTION [6 upvotes]: 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?

REPLY [7 votes]: $$\mu_n = \sum_{i=1}^n \frac{1}{n} \delta_{\frac{i}{n}}$$