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Subset Generators for Nonsingular Transformations
Author(s) -
Ellis Martin H.,
Friedman Nathaniel A.
Publication year - 1979
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-20.2.347
Subject(s) - invertible matrix , countable set , mathematics , separable space , partition (number theory) , combinatorics , group (periodic table) , discrete mathematics , pure mathematics , mathematical analysis , physics , quantum mechanics
Let ( X, B ,m ) be a separable probability space, and let { T s : s ∈ s } be an infinite collection of invertible bimeasurable nonsingular transformations of X onto X . A sufficient condition is given for the existence of a countable partition P of X for which the class { T s p : p ∈ P , s ∈ s } generates B (mod m ). This condition is satisfied by every infinite subcollection of every freely acting group of transformations. Also, sufficient (and sometimes necessary) conditions are given for the existence of a set A in B for which { T s A : s∈|S } is dense in B .