Let H be a compact subgroup of a locally compact group G. We consider the homogeneous space G/H equipped with a strongly quasi-invariant Radon measure μ. For 1≤p≤+∞, we introduce a norm decreasing linear map from L^p(G) onto L^p(G/H,μ) and show that L^p(G/H,μ) may be identified with a quotient space of L^p(G). Also, we prove that L^p(G/H,μ) is isometrically isomorphic to a closed subspace of L^p(G). These help us study the structure of the classical Banach spaces constructed on a homogeneous space via those created on topological groups.