  Let $G$ be a locally compact group which is $\sigma $-compact, endowed with a
left Haar measure $\lambda .$ Denote by $e$ the unit element of $G$, and by $B$
an open relatively compact and symmetric neighbourhood of $e$. For every $(p,q)
$ belonging to $[ 1 ; +\infty ] ^{2}$, we give an equivalent and a priori more
manageable definition of the Banach space $L_{(q,p)}^{\pi}(G),$ defined by R.
C. Busby and H. A. Smith in \cite% {1}. In the case $G$ is a group of
homogeneous type, we look at the subspaces $(L^{q},L^{p}) ^{\alpha}(G)$ of the
space $% L_{(q,p)}^{\pi}(G)$. Theses subspaces are extensions to non abelian
groups of the spaces of functions with integrable mean, defined by I. Fofana in
\cite{2}. Finally we show that $L^{\alpha ,+\infty}(G)$ is a complex subspace
of $(L^{q},L^{p}) ^{\alpha}(G)$.
