Weak convergence of a measure on a network to the Lebesgue measure

Question

Let $Z$ be a lattice of $\mathbb{R}^d$ (with $d\in\mathbb{N}^*$), and $\mu$ be a measure with support $Z$, which gives the same positive weight to every points of $Z$. For $n\in\mathbb{N}^*$, let $u_n$ be the homothety of $\mathbb{R}^d$ with ratio $(1/n)$, and $\mu_n$ be the image measure of $(n^{-d}\mu)$ by $u_n$. How can we show rigorously that $\mu_n$ converges to the Lebesgue measure of $\mathbb{R}^d$ (multiplied by a positive coefficient)?