I am looking at some brief introduction to Haar measures and since I'm not understanding basic notion, I would greatly appreciate any clarification.
Let $G$ be a locally compact group and say we have a left Haar measure $dg$. This I think I understand: if we have a suitable function $f : G \rightarrow \mathbb{C}$ then we can compute $$ \int_G f(g) dg. $$ But I don't quite understand what they mean when they write $d(hg)$ (for some $h \in G$) or $d(g^{-1})$... Any clarification would be appreciated. Thank you.
If $\lambda$ is some measure on $G$, $i : G \to G$ is the inverse map $i(g) = g^{-1}$ and $L_h : G \to G$ is the left translation $L_h (g) = hg$, one may consider the measures defined by $\lambda_{-1} (A) = \lambda (i(A))$ and $\lambda_h (A) = \lambda (L_h (A))$ for all Borel subsets $A \subseteq G$. (The maps $i$ and $L_h$ indeed take Borel subsets to Borel subsets because they are homeomorphisms.) If you know what the push-forward of a measure is, then $\lambda_{-1} = (i^{-1})_* \lambda$ and $\lambda_{h} = (L_h ^{-1})_* \lambda$. The notations $\mathrm d (g^{-1})$ and $\mathrm d (hg)$ refer precisely to these measures. More precisely, they are notations for
$$\int _G f(g) \ \mathrm d (g^{-1}) = \int _G f \ \mathrm d \lambda _{-1} = \int _G f \ \mathrm d (i^{-1})_* \lambda = \int _G f \circ i \ \mathrm d \lambda = \int _G f (g^{-1}) \ \mathrm d g$$
and
$$\int _G f(g) \ \mathrm d (h g) = \int _G f \ \mathrm d \lambda _h = \int _G f \ \mathrm d (L_h^{-1})_* \lambda = \int _G f \circ L_h \ \mathrm d \lambda = \int _G f (h g) \ \mathrm d g \ .$$
Now, if $\lambda$ is some left-invariant Haar measure, then $\lambda = \lambda_h$, i.e. $\mathrm d g = \mathrm d (hg)$ for all $h \in G$. If, furthermore, $G$ is unimodular (in particular, if it is compact or commutative), then $\lambda = \lambda_{-1}$, i.e. $\mathrm d g = \mathrm d (g^{-1})$.