Suppose $G$ is a compact Lie group, and $\omega$ is a left-invariant volume form over $G$ such that.
$$\int_{G}\omega = 1$$
with respect to the orientation determined by $\omega$. If I define
$$\int_{G}f(g)dg:=\int_{G}f\omega$$
for every continuous real-valued function $f\colon G\to \mathbb{R}$, the map $f\mapsto\int_{G}f(g)dg$ is called the Haar integral. I need to prove that $$\int_{G}f(g^{-1})dg=\int_{G}f(g)dg$$ for any continuous function $f$. I have made some attempts without success. First, I tried to use the modular function $c\colon G\to \mathbb{R}$ given by $R_{g}^{*}\omega = c(g)\omega$ for every $g\in G$. I've seen that, calling the inversion map $\beta$, we have $$L_{g}^{*}\beta^{*}(\omega)=(\beta \circ L_{g})^{*}\omega=(R_{g^{-1}} \circ \beta)^{*}\omega=\beta^{*}(c(g^{-1})\omega)=c(g^{-1})\beta^{*}\omega$$ so $\beta^{*}\omega$ might not be necessarily left invariant (for example. if $G$ is not connected). With a similar argument, I managed to prove that $\beta^{*}\omega$ is right invariant (but didn't see how to use that to my advantage). Another idea I had was to use the fact that the Haar integral is left and right invariant (which I was able to prove beforehand). Again, I don't see a clear way to get from that to the desired equality. Are any of my ideas on the right track?
The book I'm reading on the subject (''Compact Lie groups", by M. R. Sepanski) says that the argument used to prove the equation is similar to that of proving
$$\int_{G}f(gh)dg=\int_{G}f(g)dg$$ for any $h\in G$.
EDIT: I've seen written that this result can be used by proving uniqueness of the integral, i.e., that if the Haar integral can be determined by the properties I already proved, and the functional
$$J(f)=\int_{G}f(g^{-1})dg$$
verifies those properties, then $J$ is the Haar integral. I don't know how to prove either of those claims though.
Thank you in advance!
Hint: Compact groups are unimodular, i.e. the modular function is identically one. To see this, we can start with the definition of the modular function $$\mu_l(Bg)=c(g)\mu_l(B)$$ where $\mu_r$ is the left Haar measure and $B$ is a borel set of finite measure. It follows from this definition that $c:G\to\mathbb{R_+}$ is continuous (via some computation), positive (since $\mu_l$ is positive by definition), and a group homomorphism onto the positive reals with multiplication.
Furthermore, if $G$ is compact, then so is the image of $c$, which uniquely determines $c=1$. This in particular implies that $\mu_l$ is also right invariant (as a measure; the volume form may differ by a sign).