Pushforward measures on homogeneous spaces of non-unimodular groups and properties of modular functions

This paper shows that a formula for the pushforward measures on the fiber bundle $G \to G / H$ of homogeneous space of locally compact groups $G$ and $H$ can be written down by using the modular functions. As a result, we obtain the equality \begin{align} \frac{ \Delta_G ( h ) }{ \Delta_{ G / N } ( h N ) } = \frac{ \Delta_H ( h ) }{ \Delta_{ H / N } ( h N ) } \end{align} on the modular functions for any closed subgroups $N$ and $H$ of a locally compact group $G$ with $N \lhd G$ and $N \subset H \subset G$ and any $h \in H$.