Modular function of a locally compact group

Assume AC, let $G$ be an LCH group and fix a left Haar measure $\mu$ on $G$ (Left Haar integral and left Haar measure). By Right translation scales left Haar measure there is, for every $g\in G$, exactly one positive real number $c(g)$ with ${\int }_{G}f(xg)d\mu (x)=c(g){\int }_{G}fd\mu (x)(f\in C_c(G))\mathrm{.}$ The modular function (also modulus) of $G$ is the function ${\mathrm{\Delta }}_G:G⟶{\mathbb{R}}_{>0},{\mathrm{\Delta }}_G(g):=c(g^{-1}),$ characterised by the displayed identity in the equivalent form ${\int }_{G}f(x{g}^{-1})d\mu (x)={\mathrm{\Delta }}_G(g){\int }_{G}fd\mu (x)(f\in C_c(G),g\in G)\mathrm{.}$ Thus the convention fixed here is that of the source quoted below: a right translate by $g^{-1}$ scales the left Haar integral by ${\mathrm{\Delta }}_G(g)$. The companion page computes ${\mathrm{\Delta }}_G(a,b)=a^{-1}$ for the positive affine group.

• Well-definedness. The definition makes no selection: for each $g$ the scalar $c(g)$ is specified by a property that Right translation scales left Haar measure proves to hold for exactly one positive real number, and ${\mathrm{\Delta }}_G$ is then the composite of $c$ with inversion. AC enters only through the suppliers of that lemma, namely Haar existence and uniqueness of left Haar measures up to scale, and is declared as a dependency.
• Independence of the normalisation. If ${\mu }^{\mathrm{\prime }}=\lambda \mu$ with $\lambda >0$ is another left Haar measure, then multiplying both sides of the defining identity by $\lambda$ shows that $c$, hence ${\mathrm{\Delta }}_G$, is unchanged. In particular the modular function depends on the group and not on the chosen Haar measure.
• Borel-level form. The scaling identity holds for every nonnegative Borel function and every $\mu$-integrable complex function, since the translate of $\mu$ is the Radon measure $c(g)\mu$ and integrals of nonnegative Borel functions are determined by their measure. This form is used throughout the page.
• Convention comparison. Sources working with right Haar measures obtain the reciprocal function; the affine example on the companion page records the resulting numerical convention ${\mathrm{\Delta }}_G(a,b)=a^{-1}$ used on this page.