$G$: Lie group of dimension $n$.
$\tilde{\Omega}$: Orientation on $G$.
$\Omega=\epsilon^1\wedge \epsilon^2\wedge \cdots \wedge \epsilon^n$ where $\epsilon^1\, \epsilon^2, \cdots ,\epsilon^n$ is positively oriented w.r.t $\tilde{\Omega}$ such that $L_g^*\epsilon^i=\epsilon^i$ $\forall $ $i$ $\forall g,h\in G$ where $L_g(h)=gh$.
I want to show that, for any $f\in C^{\infty}(G)$ and for any $g\in G$, $$ \int_G L_g^* f \Omega=\int_G f\Omega.$$
I think I only need to show that $L_g$ is orientation preserving. Then the problem here is that we don't know the orientation. It seems showing that $L_g^*\Omega=\Omega$ is reasonable but $\Omega$ can be zero at some point so we cannot say that it is an orientation.
Am I thinking wrongly or there is another way to show this?
Thank you in advance!