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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)

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