Vol. 87, No. 2, 1980

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Naturally integrable functions

Lester Eli Dubins and David Samuel McIntyre Margolies

Vol. 87 (1980), No. 2, 299–312
Abstract

A bounded function f defined on an amenable group G is naturally integrable if, for every pair of left-invariant means μ and μ, μ(f) = μ(f). If G is the additive group of integers, then (i) f is naturally integrable if, and only if,

      ∑
lim n−1    f(j + i)  (1 ≦ i ≦ n )

exists uniformly in j , and (ii) the associated natural measure ν is convex; that is, for every pair of naturally measurable sets of integers E0 and E1 with E0 E1, there is a monotone family of naturally measurable sets Et (0 t 1) such that ν(Et) (0 t 1) is a closed interval. Analogous results hold for the presently known amenable groups.

Mathematical Subject Classification 2000
Primary: 43A07
Milestones
Received: 21 December 1978
Published: 1 April 1980
Authors
Lester Eli Dubins
David Samuel McIntyre Margolies