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On products of Radon measures. (English) Zbl 0936.28007
Authors’ abstract: “Let \(X= [0,1]^\Gamma\) with \(\text{card }\Gamma\geq{\mathfrak c}\) (\({\mathfrak c}\) denotes the continuum). We construct two Radon measures \(\mu\), \(\nu\) on \(X\) such that there exist open subsets of \(X\times X\) which are not measurable for the simple outer product measure. Moreover, these measures are strikingly similar to the Lebesgue produt measure: for every finite \(F\subseteq \Gamma\), the projections of \(\mu\) and \(\nu\) onto \([0,1]^F\) are equivalent to the \(F\)-dimensional Lebesgue measure. We generalize this construction to any compact group of weight \(\geq{\mathfrak c}\), by replacing the Lebesgue product measure with the Haar measure”.

MSC:
28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
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