Let $G$ be a compact Lie group. Furthermore, let $f$ denote throughout the question a continuous complex-valued function on $G$. Then the Haar measure on $G$ is a left-invariant measure, i.e.
$$\int_{G}\,\mathrm{dg}\,f(hg)=\int_{G}\,\mathrm{dg}\,f(g)$$
for all $h\in G$. First of all, I would like to ask if the Haar measure is also invariant under inversion, i.e. is it true that
$$\int_{G}\,\mathrm{dg}\,f(g^{-1})=\int_{G}\,\mathrm{dg}\,f(g)$$
Furthermore, one can introduce a $\delta$-distribution on $G$, namely by defining
$$\int_{G}\,\mathrm{dg}\,\delta(gh^{-1})f(g)=f(h)$$
Is it then true that
$$\int_{G}\,\mathrm{dg}\,\delta(hg^{-1})f(g)=\int_{G}\,\mathrm{dg}\,\delta(g^{-1})f(h^{-1}g)=\int_{G}\,\mathrm{dg}\,\delta(g)f(h^{-1}g^{-1})=f(h^{-1})?$$