Example of $\mu_n$ absolutely continuous to Lebesgue but $\mu$ is not

Question

Give an example of a sequence of probability measures $(\mu_n)_{n\in\mathbb{N}}$ on $\mathbb{R}$ equipped with Borel $\sigma$-algebra converging in weak topology to a probability measure $\mu$ such that each $\mu_n$ are absolutely continuous with respect to Lebesgue measure, but $\mu$ is not.

I thought of using $\delta_{\frac{1}{n}}$ once it would converge to $\delta_0$ which implies $\delta_0{0}=1$. But then each $\delta_{\frac{1}{n}}$ is not absolutely continuous to Lebesgue once $\lambda(\frac{1}{n})=0$ for all $n$. I do not know a lot of probability to pick up any more examples.

Question:

Which could be an example of a probability measure with the aforementioned conditions?

Thanks in advance!

Answer 1

Take $\mu_n$ to be the uniform distribution on $[0,1/n]$. Then each $\mu_n$ is absolutely continuous with respect to Lebesgue's measure, but the sequence converges weakly to $\delta_0$ which is not.

This is easily seen by the fact that the distribution function of $\mu_n$ converges to $1_{[0,\infty)}$ (the distribution function of $\delta_0$) in all of its continuity points.