  In the first part, we construct a cut and project scheme from a family
$\{P_\varepsilon\}$ of sets verifying four conditions. We use this construction
to characterize weighted Dirac combs defined by cut and project schemes and by
continuous functions on the internal groups in terms of almost periodicity. We
are also able to characterise those weighted Dirac combs for which the internal
function is compactly supported. Lastly, using the same cut and project
construction for $\varepsilon$-dual sets, we are able to characterise Meyer
sets in $\sigma$-compact locally compact Abelian groups.
