Let $a_1, a_2, \ldots$ be any sequence of non-negative real numbers with $\sum_i a_i = 1$. Define the discrete measure $\mu$ by $\mu(\cdot) = \sum_{i\in\mathbb{N}} a_i \delta_i(\cdot)$, where $\delta_i(\cdot)$ is a point-mass at the positive integer $i$. Construct a sequence $\{\mu_n\}$ of probability measures, each having a density with respect to the Lebesgue measure, such that $\mu_n\Rightarrow \mu$.
Note: $\mu_n\Rightarrow \mu$ denotes the weak convergence of $\mu_n$ to $\mu$.
Official solution:
My problem is that I always thought that density functions have to be continuous, but that doesn't seem to be the case here, or does it? So is the official solution correct?
Or what would be an example of continuous density functions $f_n$ for which the measures $\mu_n(A) = \int_A f_n \, d\lambda$ converge weakly to $\mu$? Should I use "Gaussian peaks"?
