  In this work we consider translation-bounded measures over a locally compact
Abelian group $\mathbb{G}$, with particular interest for their so-called
diffraction. Given such a measure $\Lambda$, its diffraction $\widehat{\gamma}$
is another measure on the Pontryagin dual $\widehat{\mathbb{G}}$, whose
decomposition into the sum $\widehat{\gamma} = \widehat{\gamma}_{\mathrm{p}}+
\widehat{\gamma}_{\mathrm{c}}$ of its atomic and continuous parts is central in
diffraction theory. The problem we address here is whether the above
decomposition of $\widehat{\gamma}$ lifts to $\Lambda$ itself, that is to say,
whether there exists a decomposition $\Lambda = \Lambda _{\mathrm{p}} + \Lambda
_{\mathrm{c}}$, where $\Lambda _{\mathrm{p}}$ and $\Lambda _{\mathrm{c}}$ are
translation-bounded measures having diffraction $\widehat{\gamma}_{\mathrm{p}}$
and $\widehat{\gamma}_{\mathrm{c}}$ respectively. Our main result here is the
almost sure existence, in a sense to be made precise, of such a decomposition.
It will also be proved that a certain uniqueness property holds for the above
decomposition. Next we will be interested in the situation where
translation-bounded measures are weighted Meyer sets. In this context, it will
be shown that the decomposition, whether it exists, also consists of weighted
Meyer sets. We complete this work by discussing a natural generalization of the
considered problem.
