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The Hahn-Banach theorem implies the Banach-Tarski paradox. (English) Zbl 0792.28006
M. Foreman and F. Wehrung [Fundam. Math. 138, No. 1, 13-19 (1991; preceding review)] showed that if \(G\) is a nonamenable group acting freely on a set \(\Omega\) and if \(\mu\) is a \(G\)-invariant finitely additive probability measure defined on a \(G\)-invariant subalgebra of \({\mathcal P}(\Omega)\), then \(\Omega\) admits nonmeasurable subsets with respect to \(\mu\). They announced that the second author established a paradoxical-type decomposition result under the same hypothesis. Their method relies on the Zermelo-Fraenkel axioms and the Hahn-Banach theorem, but not on the axiom of choice.
Exploiting ideas of that article, Pawlikowski, by a short subtle argument concerning a finitely additive measure on the free product of the powerset algebras of all \(F_ 2\)-orbits, proves the Banach-Tarski paradox (“duplication of the cube”): If the (nonamenable) free group \(F_ 2\) of two generators acts freely on \(X\) (i.e., \(xf\neq x\) whenever \(x\in X\) and \(f\) differs from the identity element in \(F_ 2\)), then \(X\) is \(F_ 2\)-paradoxical.

MSC:
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
03E25 Axiom of choice and related propositions
46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
43A05 Measures on groups and semigroups, etc.
28A12 Contents, measures, outer measures, capacities
51M25 Length, area and volume in real or complex geometry
Citations:
Zbl 0792.28005
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