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Let $G$ be a locally compact group which is $\sigma $-compact, endowed with aleft 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 moremanageable 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 ofhomogeneous type, we look at the subspaces $(L^{q},L^{p}) ^{\alpha}(G)$ of thespace $% L_{(q,p)}^{\pi}(G)$. Theses subspaces are extensions to non abeliangroups 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 subspaceof $(L^{q},L^{p}) ^{\alpha}(G)$.

Espaces de fonctions \`a moyenne fractionnaire int\'egrable sur les groupes localement compacts