  In presence of gravitational radiation, the notion of angular momentum of an
isolated system acquires an infinite dimensional supertranslation ambiguity.
This fact has been emphasized in the mathematical general relativity literature
over several decades. We analyze the issue in the restricted context of compact
binary coalescence (CBC) where the initial total angular momentum of the binary
and the final black hole spin generically refer to \emph{distinct} rotation
subgroups of the Bondi-Metzner-Sachs group, related by
\emph{supertranslations}. We show that this ambiguity can be quantified using
gravitational memory and the `black hole kick'. Our results imply that,
although the ambiguity is conceptually important, under assumptions normally
made in the CBC literature, it can be ignored in practice for the current and
foreseeable gravitational wave detectors.
