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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
##### Keywords:
product measure problem; Radon measure; Haar measure
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