Invariant signed measures and the cancellation law

Author:
M. Laczkovich

Journal:
Proc. Amer. Math. Soc. **111** (1991), 421-431

MSC:
Primary 28D15; Secondary 20B99

DOI:
https://doi.org/10.1090/S0002-9939-1991-1036988-2

MathSciNet review:
1036988

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a set, and let the group act on . We show that, for every , the following are equivalent: (i) and are -equidecomposable; and (ii) for every -invariant finitely additive signed measure . If the sets and the pieces of the decompositions are restricted to belong to a given -invariant field , then if and only if the cancellation law holds in the space . We show that the cancellation law may fail even if the transformation group is Abelian.

**[1]**A. H. Clifford and G. B. Preston,*The algebraic theory of semigroups*, Math. Surveys**7**, Amer. Math. Soc., 1961. MR**0132791 (24:A2627)****[2]**R. J. Gardner and M. Laczkovich,*The Banach-Tarski theorem on polygons and the cancellation law*, Proc. Amer. Math. Soc.**109**(1990), 1097-1102. MR**1017001 (90k:52001)****[3]**W. Sierpiński,*On the congruence of sets and their equivalence by finite decomposition*, Lucknow, 1954; reprinted by Chelsea, 1967. MR**0060567 (15:691c)****[4]**A. Tarski,*Über das absolute Mass linearer Punktmengen*, Fund. Math.**30**(1938), 218-234.**[5]**-,*Cardinal algebras*, Oxford Univ. Press, 1949.**[6]**J. K. Truss,*The failure of cancellation laws for equidecomposability types*, Canad. J. Math. (to appear). MR**1074225 (91k:03147)****[7]**S. Wagon,*The Banach-Tarski paradox*, Cambridge Univ. Press, 1986. MR**1251963 (94g:04005)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
28D15,
20B99

Retrieve articles in all journals with MSC: 28D15, 20B99

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1991-1036988-2

Article copyright:
© Copyright 1991
American Mathematical Society