Remark 3

— Suppose $X/H$ paracompact, and let us retain the notations of Proposition 9. The mapping $k \mapsto h \cdot (k \circ \pi)$ of $L^1(X/H, \lambda)$ into $L^1(X, \lambda^\#)$ is then isometric by (14) and is a right inverse of the mapping $f \mapsto f^b$ of $L^1(X, \lambda^\#)$ onto $L^1(X/H, \lambda)$.