Consider the sequence $(\mu_n)_{n\in \mathbb{N}}$ of Borel measures on $\mathbb{R}$ given by:
$$ \mu_n = \sum_{k\in \mathbb{Z}} \Delta x_{n,k} \, \delta_{x_{n,k}} $$ where $\delta_x$ denotes the Dirac measure at $x$ and $x_{n,k}$ is a given sequence.
I want to say that the sequence of measures $(\mu_n)$ converges to the Lebesgue measure $\lambda$ on $\mathbb{R}$. The problem is that I am not sure which is the most natural / strongest notion of convergence here.
Now some details. For simplicity let's just take $x_{n,k} = \frac{k}{n}$. I have denoted $$\Delta x_{n,k} = \frac{x_{n,k+1} - x_{n,k-1}}{2}$$ so here $\Delta x_{n,k} = \frac{1}{n}$. (More generally, I require that $(x_{n,k})_{k\in \mathbb{Z}}$ is an increasing sequence such that $\lim_{k \to \pm\infty} x_{n,k} = \pm \infty$, and that $\sup_k \Delta x_{n,k}$ converges to $0$ when $n \to \infty$.)
The sequence of measures that I am actually interested in is a higher dimensional analogue of this, but I think this 1D example captures the idea.
I figured out a neat proof that $$ \lim_{n \to \infty}\mu_n(A) \to \lambda(A) \qquad (1) $$ for any continuity set $A$ (that's a Borel set whose boundary has zero measure: $\lambda(\delta A) = 0$). EDIT: I realize thanks to the comment of Sangchul Lee that there was a problem in my proof: I also need $A$ to be bounded (or just $\delta A$ bounded would be sufficient).
If we were talking about probability measures, I believe the question is settled: we have "weak convergence" (some would say "weak-* convergence") of $\mu_n$ to $\lambda$, for any reasonable definition of "weak convergence". (See for example Convergence of measures in Wikipedia.)
But here the measures are not finite. Surely I can take "weak" (more like weak-*) convergence relative to the space of compactly supported continuous functions $\mathcal{C}_c(\mathbb{R})$. But maybe there's a stronger and more natural form of "weak" convergence to consider. Ideally, it should be equivalent to property $(1)$ with continuity sets. For example, I could try instead $\mathcal{C}^+(\mathbb{R})$: nonnegative continuous functions (allowing infinite integrals), or $\mathcal{C}_b^+(\mathbb{R})$ (bounded) to mimic "narrow convergence", or $\mathcal{C}_0^+(\mathbb{R})$ (continuous functions converging to $0$ at infinity).