Remark 2

Let us conserve the hypotheses and notations of Prop. 8. The mapping $g \mapsto h \cdot (g \circ \pi)$ is a continuous mapping of $\mathscr{K}(X/H)$ into $\mathscr{K}(X)$ that is a right inverse of the mapping $f \mapsto f^b$. Consequently, every bounded (resp. compact) subset of $\mathscr{K}(X/H)$ is the image of a bounded (resp. compact) subset of $\mathscr{K}(X)$. From this, one deduces immediately that the mapping $\lambda \mapsto \lambda^\#$ is again an isomorphism of $\mathscr{M}(X/H)$ onto a closed linear subspace of $\mathscr{M}(X)$ when these spaces are equipped with the topology of bounded (resp. compact) convergence.