Controlled Delone sets and continuous family of Roe algebras

We introduce the notion of controlled Delone sets and show that, for a proper metric space $X$ the set consisting of controlled Delone subsets and of $X$ is compact. Under some natural restrictions we show that, for any sequence of controlled Delone sets convergent to $X$, there exists a tautological continuous field of $\mathrm C^*$-algebras with the uniform Roe algebras of the Delone sets as fibers over finite points, and with the Spakula's version of the Roe algebra as the fiber over $X$.