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On systems of imprimitivity on locally compact abelian groups with dense actions

J. Mathew;  M. G. Nadkarni — 1978

Annales de l'institut Fourier

Consider the four pairs of groups ( Γ , R ) , ( Γ / Γ 0 , R / Γ 0 ) , ( K ∩ S , P ) and ( S , B ) , where Γ , R are locally compact second countable abelian groups, Γ is a dense subgroup of R with inclusion map from Γ to R continuous; Γ 0 ⊆ Γ ⊆ R is a closed subgroup of R ; S , B are the duals of R and Γ respectively, and K is the annihilator of Γ 0 in B . Let the first co-ordinate of each pair act on the second by translation. We connect, by a commutative diagram, the systems of imprimitivity which arise in a natural fashion on each pair, starting with a system...

Sets with doubleton sections, good sets and ergodic theory

A. Kłopotowski;  M. G. Nadkarni;  H. Sarbadhikari;  S. M. Srivastava — 2002

Fundamenta Mathematicae

A Borel subset of the unit square whose vertical and horizontal sections are two-point sets admits a natural group action. We exploit this to discuss some questions about Borel subsets of the unit square on which every function is a sum of functions of the coordinates. Connection with probability measures with prescribed marginals and some function algebra questions is discussed.

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