On the existence of a Haar measure in topological IP-loops. (English) Zbl 1242.28022

Let \(G\times G\rightarrow G\) be a binary (not necessarily associative) operation on a set G such that \((G,\cdot)\) has a a neutral element and for every \(x\in G\) there is an element \(x^{-1}\) with \((yx)x^{-1}=y=x^{-1}(xy)\) for any \(y\in G\). Then \((G,\cdot)\) is called an IP-loop. A topological IP-loop is an IP-loop endowed with a topology such that \((x,y)\mapsto x^{-1}y\) is continuous. The authors prove that in every locally compact topological IP-loop whose topology is induced by a left-invariant uniformity there exists at least one left Haar measure, i.e., a left-invariant non-zero \([0,\infty]\)-valued \(\sigma\)-additive measure defined on the \(\sigma\)-algebra generated by the compact subsets of \(G\) such that any compact subset of \(G\) has finite measure. As mentioned by the authors, they proved in another paper that a Haar measure on a locally compact topological IP-loop unique up to a factor.
Reviewer: Hans Weber (Udine)


28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
20N05 Loops, quasigroups
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