  In this paper we introduce the concept of Wiener-Luxemburg amalgam spaces
which are a modification of the more classical Wiener amalgam spaces intended
to address some of the shortcomings the latter face in the context of
rearrangement-invariant Banach function spaces.
  We introduce the Wiener-Luxemburg amalgam spaces and study their properties,
including (but nor limited to) their normability, embeddings between them and
their associate spaces. We also study amalgams of quasi-Banach function spaces
and introduce a necessary generalisation of the concept of associate spaces. We
then apply this general theory to resolve the question whether the
Hardy-Littlewood-P\'{o}lya principle holds for all r.i. quasi-Banach function
spaces. Finally, we illustrate the asserted shortcomings of Wiener amalgam
spaces by providing counterexamples to certain properties of Banach function
spaces as well as rearrangement invariance.
