  Given a bounding class $B$, we construct a bounded refinement $BK(-)$ of
Quillen's $K$-theory functor from rings to spaces. $BK(-)$ is a functor from
weighted rings to spaces, and is equipped with a comparison map $BK \to K$
induced by "forgetting control". In contrast to the situation with $B$-bounded
cohomology, there is a functorial splitting $BK(-) \simeq K(-) \times
BK^{rel}(-)$ where $BK^{rel}(-)$ is the homotopy fiber of the comparison map.
