This paper initiates the systematic study of the preservation of quasi-uniform properties between a quasi-uniformity

$\mathcal{U}$ on a set

$X$ and the Bourbaki quasi-uniformity

${\mathcal{U}}_{*}$ on the collection

${\mathcal{P}}_{0}\left(X\right)$ of all nonempty subsets of

$X$. The authors prove that

$({\mathcal{P}}_{0}\left(X\right),{\mathcal{U}}_{*})$ is precompact (totally bounded) if, and only if,

$(X,\mathcal{U})$ is precompact (totally bounded), and they give examples to show that the corresponding results hold neither for compactness nor hereditary precompactness. The principal result is an extension of the Isbell-Burdick Theorem: The Bourbaki quasi-uniformity

${\mathcal{U}}_{*}$ is right K-complete if, and only if, each stable filter on

$(X,\mathcal{U})$ has a cluster point. As might be expected, along the way the authors provide a good many interesting results and examples concerning both right K-completeness and the related property that each stable filter has a cluster point.