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.

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    $\begingroup$ To integrate continuous functions it suffices to know how to integrate piecewise constant functions $f(g) = \sum_j c_j 1_{g\in A_j}$. Then $\int_G 1_{g \in A} dg = \mu(A)$ where $\mu$ is the left Haar measure satisfying $0 \le \mu(A) = \mu(g A),A \cap B = \emptyset, \mu(A \cup B) = \mu(A)+\mu(B)$. And for $\phi,\phi^{-1}$ both continuous $G \to G$, $\int_G 1_{g \in A} d\phi(g) =\int_{\phi(G)} 1_{\phi^{-1}(x) \in A} d\phi(\phi^{-1}(x))=\int_G 1_{x\in \phi(A)} dx= \mu(\phi(A))$ $\endgroup$ – reuns Apr 17 '19 at 15:19

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

$$\int _G f(g) \ \mathrm d (g^{-1}) = \int _G f \ \mathrm d \lambda _{-1} = \int _G f \ \mathrm d (i^{-1})_* \lambda = \int _G f \circ i \ \mathrm d \lambda = \int _G f (g^{-1}) \ \mathrm d g$$


$$\int _G f(g) \ \mathrm d (h g) = \int _G f \ \mathrm d \lambda _h = \int _G f \ \mathrm d (L_h^{-1})_* \lambda = \int _G f \circ L_h \ \mathrm d \lambda = \int _G f (h g) \ \mathrm d g \ .$$

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. $\mathrm d g = \mathrm d (g^{-1})$.

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