Let $N_n$ be a sequence of natural numbers increasing to infinity, and suppose we have a sequence of finite sets of distinct points $X_n = \{x_1^{n},x_2^{n},\ldots,x^{n}_{N_n}\} \subset[0,1] \subset \mathbb{R}$. Consider the discrete probability measure $$ \rho_n = \frac{1}{N_n}\sum_{i=1}^{N_n}\delta_{x^{n}_i}, $$ a normalized sum of delta functions centered at the points $x^{n}_i$. Being bounded as a linear operator on $C([0,1])$, there exists a vaguely convergent subsequence of the $\rho_n$ i.e. there exists a probability measure $\rho$ on [0,1] such that $$ \int_0^1fd\rho_{n_k} \to_{k\to \infty} \int_0^1 fd\rho $$ for all $f \in C([0,1])$. Let me further impose a spacing condition that if $$ r_n := \inf_{i\neq j} |x^n_i - x^n_j| $$ is the minimum distance between distinct pairs of the $x^n_i$, then $$ \inf_n N_n r_n > 0. $$ (in particular, this implies $x^n_i$ are distinct). This loosely can be interpreted as enforcing that the $X_n$ not accumulate too much on 0-dimensional sets (or perhaps I should say on sets of Hausdorff/Minkowski dimension < 1? I'm not sure and would be interested in answers to this as well, though it's not my main question).

As a simple example, if $X_n$ is regularly spaced on $[0,1]$, then $d\rho = dx = $ Lebes**g**ue measure.

My question is: What further conditions can be imposed on the sets $X_n$ to guarantee that the original sequence $\rho_n$ converges (as opposed to a subsequence)?

Note that this is a rewrite of an earlier question of mine (my first ever), since closed (Uniqueness of the limit of a sequence of (discrete) probability measures). I understand if it gets closed again, because it's pretty specific yet open-ended at the same time. I imagine any nontrivial answer would be kind of creative, perhaps involving a rule or algorithm for how the points in the $X_n$ are distributed, and/or involving some nestedness property. Nestedness alone (i.e. $X_n \subset X_{n+1}$) does not guarantee uniqueness of the limit, as I have constructed counterexamples to demonstrate.