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 $G\to G\mathrm{/}H$ 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{ Δ_G ( h ) }{ Δ_{ G / N } ( h N ) } = \frac{ Δ_H ( h ) }{ Δ_{ H / N } ( h N ) } \end{align} on the modular functions for any closed subgroups $N$ and $H$ of a locally compact group $G$ with $N⊲G$ and $N\subset H\subset G$ and any $h\in H$.