# Some notational question related to Haar measure (what do $d g^{-1}$ or $d (hg)$ mean?)

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.

• 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 $G \to G$, $\int_G 1_{g \in A} d\phi(g) =\int_{\phi(G)} 1_{\phi^{-1}(x) \in A} d\phi(\phi^{-1}(x))=\int_G 1_{x\in \phi(A)} dx= \mu(\phi(A))$ – 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. $$\mathrm d g = \mathrm d (g^{-1})$$.