  In this paper, we prove that the inclusions between Morrey spaces, between
weak Morrey spaces, and between a Morrey space and a weak Morrey space are all
proper. The proper inclusion between a Morrey space and a weak Morrey space is
established via the unboundedness of the Hardy-Littlewood maximal operator on
Morrey spaces of exponent 1. In addition, we also give a necessary condition
for each inclusion. Our results refine previous inclusion properties studied in
[Gunawan et al, \emph{Math. Nachr.} {\bf 290} (2017), 332--340].
