The use of shears to construct paradoxes in 𝑅²

Wagon, Stanley

doi:10.1090/S0002-9939-1982-0656101-5

The use of shears to construct paradoxes in $\textbf {R}^{2}$
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Proc. Amer. Math. Soc. 85 (1982), 353-359 Request permission

Abstract:

It is shown that the addition of a certain shear transformation to the planar isometry group is sufficient to allow a Banach-Tarski type paradox to be constructed in ${{\mathbf {R}}^2}$. This paradox is then combined with a result of Rosenblatt to obtain a characterization of two-dimensional Lebesgue measure as a finitely additive measure.

References

Sur le problème de la mésure

4

Un théorème sur les transformations biunivoques

6

Sur la décomposition des ensembles de points en parties respectivement congruentes

6

Joël Lee Brenner, Quelques groupes libres de matrices, C. R. Acad. Sci. Paris 241 (1955), 1689–1691 (French). MR 75952
K. Goldberg and M. Newman, Pairs of matrices of order two which generate free groups, Illinois J. Math. 1 (1957), 446–448. MR 89207
Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0251549
F. Hausdorff, Bemerkung über den Inhalt von Punktmengen, Math. Ann. 75 (1914), no. 3, 428–433 (German). MR 1511802, DOI 10.1007/BF01563735

Theorie der elliptischen Modulfunktionen

A. G. Kurosh, The theory of groups. Vol. II, Chelsea Publishing Co., New York, N.Y., 1956. Translated from the Russian and edited by K. A. Hirsch. MR 0080089

Leçons sur l’intégration

G. A. Margulis, Some remarks on invariant means, Monatsh. Math. 90 (1980), no. 3, 233–235. MR 596890, DOI 10.1007/BF01295368

Finitely additive invariant measures on Euclidean spaces

Jan Mycielski, Finitely additive invariant measures. I, Colloq. Math. 42 (1979), 309–318. MR 567569, DOI 10.4064/cm-42-1-309-318

Zur allgemeinen Theorie des Masses

13

Joseph Rosenblatt, Uniqueness of invariant means for measure-preserving transformations, Trans. Amer. Math. Soc. 265 (1981), no. 2, 623–636. MR 610970, DOI 10.1090/S0002-9947-1981-0610970-7
I. N. Sanov, A property of a representation of a free group, Doklady Akad. Nauk SSSR (N. S.) 57 (1947), 657–659 (Russian). MR 0022557
Wacław Sierpiński, Sur l’équivalence des ensembles par décomposition en deux parties, Fund. Math. 35 (1948), 151–158 (French). MR 28377, DOI 10.4064/fm-35-1-151-158
Karl Stromberg, The Banach-Tarski paradox, Amer. Math. Monthly 86 (1979), no. 3, 151–161. MR 522339, DOI 10.2307/2321514
Dennis Sullivan, For $n>3$ there is only one finitely additive rotationally invariant measure on the $n$-sphere defined on all Lebesgue measurable subsets, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 1, 121–123. MR 590825, DOI 10.1090/S0273-0979-1981-14880-1

Problème

7

Stanley Wagon, Circle-squaring in the twentieth century, Math. Intelligencer 3 (1980/81), no. 4, 176–181. MR 642135, DOI 10.1007/BF03022979