In the famous paper by Benjamini-Schramm 2001, they consider the space of rooted graphs with uniformly bounded degrees, this space modulo rooted graph isomorphism is denoted by $\mathcal X$.
This space is equiped with a local metric: the distance between two rooted graphs $(G,o)$ and $(G',o')$ is $2^{-k}$, if $k$ is the maximal integer such that the $k$-ball of the root in $G$ and $G'$ are isomorphic as rooted graphs.
It follows that $\mathcal X$ is a compact metric space. A random rooted graph is a borel probablistic measure on $\mathcal X$. In the paper, they define the so-called Benjamini-Schramm convergence of a sequence of random rooted graphs $(G_n,o_n)\rightarrow (G,o)$ if and only if, for any finite rooted graph $(H,o')$ and $k\geqslant 0$, the probability that $(H,o')$ is rooted isomorphic to the $k$-neighborhood of $o_n\in G_n$ converges to the probability that $(H,o')$ is rooted isomorphic to the $k$-neighborhood of $o\in G$.
It seems to me that their definition of convergence amounts to saying that the measure of any metric balls of $\mathcal X$ converges. But the general definition of weak convergence requires more than that. One of the equivalent definitions write: for any bounded continuous function $f$ on $\mathcal X$, the integral pairing converges: $$\langle f,(G_n,o_n)\rangle\rightarrow \langle f,(G,o)\rangle.$$
My question is whether the Benjamini-Schramm convergence is equivalent to the weak convergence of Borel measure?
Edit: thank Will for pointing out a typo.
p.s. In the present situation, every metric ball is open and closed. Hence the characteristic function of any metric ball of $\mathcal X$ is continuous. So weak convergence would imply Benjamini-Schramm convergence. I want to figure out the reverse direction.
