Let the measures of locally compact groups $\,K < G\,$ be $\, dk\,$ and $\, dg\,$, correspondingly. For a Hilbert space $\mathbb{V}$ equipped with a dot product $\,\langle~,~\rangle\,$, introduce the space ${{\cal{L}}^G}$ of square-integrable functions: $$ {\cal{L}}^G=\left\{\right.\varphi\,:\, G\longrightarrow{\mathbb{V}}\; {\LARGE\mbox{$|$}}\int dg\,\langle \varphi(g),\varphi^{\,\prime}(g)\rangle<\infty\;\;\, \mbox{for}\;\forall\,\varphi,\,\varphi^{\,\prime} \left.\right\}. $$
Of interest to us will be the subspace of Mackey functions, those obeying the equivariance condition $$ \varphi(gk)=D^{-1}(k)\,\varphi(g)~,~~g\in G\,,~~k\in K~, $$ where a representation $\, D(K)\,$ is assumed (pseudo)unitary, i.e. preserving $\,\langle~,~\rangle\,$.
Treat the group $G$ as a principal bundle on the base $G/K$, and fix a section $x=\gamma(X)$, with each $x$ acting as the representative of a coset $X=xK$.
Functions on cosets can be naturally introduced as functions on the section: $$ \tilde{\varphi}(X)\equiv\varphi(x)~. $$
We also can define a dot product on the section as $$ \int \langle\, \varphi(x)\, ,\,\varphi^{\,\prime\,}(x)\,\rangle\,dx~.\qquad\qquad\qquad(*) $$
To be able to interpret it as a dot product on the quotient space, we must be sure that it remains invariant under a different section choice, say $\bar{x}=\bar{\gamma}(X)$. The two sections are related via $x\,=\,\bar{x}\, k(x)$, where $k(x)\in K$. Plugging this in the above expression, and keeping in mind that $\varphi(x)=\varphi(\bar{x}\, k(x))= D^{-1}(\, k(x)\,)\,\varphi(\bar{x})\,$, we obtain: $$ \int\langle\,\varphi(\bar x)\,,\,\varphi(\bar x)\,\rangle\; d\bar x~=~ \int\langle\varphi(x)\,,\,\varphi(x)\rangle\; d (\, x\, k(x)\,)~. $$
QUESTION 1. $~~$I have been sloppy by writing simply $k(x)$ instead of $k(X)$ where $X\in G/K$. Is this forgivable for a right-invariant measure $dg\,$?
QUESTION 2. $~~$Assuming the measure is right-invariant, can we say that $ d (\, x\, k(x)\,)=dx $ for any reasonable function $k(x)$?
/"Reasonable" $k(x)$ means: "good enough for a physicist", so that the representatives $x$ always make Borel algebras and the measure $dx$ is well defined./
QUESTION 3. $~~$Was the assumption of (pseudo)unitarity of $D(K)$ inded necessary here?
QUESTION 4. $~~$If the answer to all the above questions happens to be affirmative, may I assume that I have proven the invariance of product (*) under a change of section, so it can be understood as a dot product in the quotient space?