Easy examples of discrete/atomic probability measures on $\mathbb{R}$ weakly converging to a probability measure with continuous density function

Question

I'm trying to construct some easy examples of sequence of discrete/atomic probability measures $P_n$ on the real line that converges weakly to a measure $P$ on the same, but that has a continuous density function $f$ (i.e. Radon-Nikodym derivative) w.r.t. the Lebesgue measure on the real line. I'm thinking along the line of Riemann integral...some hints of solutions would be appreciated!

Answer 1

Take the sequence of $\frac{\delta_{0}+\delta_{\frac{1}{n}}+....+\delta_{\frac{n-1}{n}}}{n}$ where each $\delta_{\frac{k}{n}}$ is a dirac measure.

If you examine the convergence with respect to continuous functions with compact support you will see that the weak limit is the measure $v(A)=m(A)$ if $A \subseteq [0,1]$ and $v(A)=0$ otherwise.

Here $m$ denotes the Lebesgue measure.