Doubling property for the Haar measure on quasi-metric groups (Q2767712)

\((X,d,\mu)\) is a space of homogeneous type if \(d\) is a non-negative symmetric function on \(X\times X\) satisfying the quasi-triangle inequality \(K(d(x,y)+ d(y,z))\geq d(x,z)\) for some \(K\geq 1\) and all \(x,y,z\) in \(X\) and if \(\mu\) is a doubling measure defined on a \(\sigma\)-algebra containing the \(d\)-balls, i.e NEWLINE\[NEWLINE\infty> A\mu\bigl( B(x,r)\bigr) \geq\mu\bigl( B(x,2r)\bigr) >0NEWLINE\]NEWLINE for some constant \(A\), every \(x\) and \(r>0\). A complete metric space has the weak homogeneity property provided there is a constant \(N=N(r)\) such that any subset \(A\) of a ball \(B(x,r)\) with the property that if \(d(y,z)\geq r/2\), for \(y\) and \(z\) in \(A\) and \(y\neq z\), then \(A\) consists of at most \(N\) points.NEWLINENEWLINENEWLINEIn this note the authors consider the doubling property for Haar measures on groups. They show that for a complete abelian group \(X\), whose topology is given by a translation invariant quasi-distance having the weak homogeneity property, then \((X,d,\mu)\) is a space of homogeneous type for \(\mu\) the Haar measure on \(X\). This provides another solution to a question of Dyn'kin [in: \textit{V. P. Khavin} (ed.) and \textit{N. K. Nikolskij} (ed.), Linear and complex analysis. Problem book, Lect. Notes Math. 1043, Berlin etc. (1984; Zbl 0545.30038)].