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Haar null sets in the space of automorphisms on \([0,1]\). (English) Zbl 0945.28009

A set \(A\subset G\) of a (non-Abelian) Polish group is called Haar null (left Haar null, right Haar null, left-and-right Haar null) if there is a Borel probability measure \(\mu \) on \(G\) such that \(\mu (hAg)=0\) (\(\mu (hA)=0\), \(\mu (Ag)=0\), \(\mu (hA)=\mu (Ag)=0\), resp.) for all \(g,h \in G\). Since in case of locally compact Polish group these concepts coincide the authors study the situation in the non-abelian non-locally compact group \(H\) of all homeomorphisms \(h:[0,1]\rightarrow [0,1]\) with \(h(0)=0\) and \(h(1)=1\). They showed that there are uncountably many disjoint non-Haar null sets in \(H\) each of them being a left-and-right Haar null set. They also give an explicit example of a measure \(\mu \) witnessing that a set \(A\) is left-and-right Haar null (right Haar null) and still \(\mu \) is not the measure witnessing that \(A\) is Haar null (left Haar null, resp.).

MSC:

28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
26A30 Singular functions, Cantor functions, functions with other special properties
43A05 Measures on groups and semigroups, etc.
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