Preliminaries: Let be $G\subset\operatorname{GL}(n,\mathbb R)$ a closed subgroup of the general linear group over $\mathbb R$. Especially, $G$ is a Lie group by the closed subgroup theorem and we set $k:=\dim(G)$. Let be $x\in G$, then there is a neigborhood $U\subset G$ of $x$ and a local chart $\phi:U\to V$ with $V\subset\mathbb R^k$. Furthermore, $G$ admits a Haar measure $\mu$ and $\lambda$ denotes the Lebesgue measure on $\mathbb R^k$. We write $\lambda\circ\phi$ for the pushforward measure of the lebesgue measure regarding $\psi^{-1}$.
Question: Is the following statement correct and does somebody know a reference to a rigorous proof for it?
There is a continuous non-vanishing density $f:U\to\mathbb R$ such that $$\mu(A) = \int_A f \mathrm d(\lambda\circ\phi)$$ for all measurable $A\subset U$.
I was looking for some literature for a while now but I only found a passage in Follands Course in Abstract Harmonic Analysis (above proposition 2.21) that states: [...] one sees that Haar measure on a Lie group is given by a smooth density times Lebesgue measure in any local coordinates [...] among some really rough guidance how to prove this using some non-basic concepts of differential geometry. I'm not really familiar with differential geometry and therefore, I'm not capable to reconstruct Follands ideas. Besides, I'm also not very interested to dive deep into differential geometry at the moment but the question seems so natural to me which makes me believe that somebody already proved it rigorously.