I'm reading the book "Basic Lie Theory" (http://guests.mpim-bonn.mpg.de/abbaspou/Lie-Book_verrouille.pdf) and I'm trying to understand the proof of Lemma 2.3.4 which states that:
Let $G$ be a locally compact group and $H$ be a closed subgroup, $\Delta_G$ and $\Delta_H$ be the Haar measure of $G$ and $H$ respectively and $I(f)(g) := \int_H f(gh)dh$. Suppose $\Delta_G |_H = \Delta_H$. Let $f \in C_0(G)$. If $I(f) = 0$, then $\int_G f(g)dg = 0$.
Most of the proof are alright except I don't understand where the equality \begin{equation} \int_G f(g) \int_H \phi(gh^{-1})\Delta_G(h)dhdg = \int_G f(g) \int_H \phi(gh) \Delta_H(h^{-1})\Delta_G(h) dhdg \end{equation} came from. I feel that the RHS should be equals to $\int_G f(g) \int_H \phi(g) dhdg$ which doesn't look like the LHS. Any help is highly appreciated.