Corollary

— Let f be a function defined on $\Omega$, with values in a Banach space or in $\overline{\mathbf{R}}$. For f to be $\mu$-integrable, it is necessary and sufficient that the function $(x, y) \mapsto f(xy) \Delta_G(y) \Delta_Y(y)^{-1}$ be $(\mu_X \otimes \mu_Y)$-integrable, in which case

where $a$ is a constant $> 0$ independent of f.

By Prop. 13, and Ch. V, §4, No. 4, Th. 2, for f to be $\mu$-integrable, it is necessary and sufficient that the function $(x, y) \mapsto f(xy^{-1})$ be integrable for $\mu_X \otimes (\Lambda^{-1} \cdot \mu_Y)$, or again that the function $(x, y) \mapsto f(xy^{-1}) \Delta_G(y)^{-1}$ be integrable for $\mu_X \otimes \mu_Y$, or again that the function $(x, y) \mapsto f(xy) \Delta_G(y) \Delta_Y(y)^{-1}$ be integrable for $\mu_X \otimes \mu_Y$. Formula (23) results from an analogous argument.