Vol. 104, No. 2, 1983

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Applications of differentiation of p-functions to semilattices

P. H. Maserick

Vol. 104 (1983), No. 2, 417–427

Let S be a commutative semigroup with identity 1 such that x2 = x for each x S (i.e. S is a semilattice). Let Γ denote the set of semicharacters equipped with topology of simple convergence and μ be a fixed probability measure on Γ. Those real-valued functions f on S which admit disintegrations of the form f(x) = Γ ρ(x)f(ρ) where either f = fwith f′∈ Lp(μ) (1 p ≤∞) or μf is singular with respect to μ, are characterized. This extends the previous characterization of Alo and Korvin from the case where p is either 1 or to all p [1,]. Applications of this theory to the classical Lp-spaces on the n-cube are also presented. The main applications occur upon specializing to the case where S is a Boolean algebra and the functions on S that are being disintegrated are additive. Not only is the Darst decomposition theorem easily recovered, but also the theory of V p-spaces of set functions introduced by Bochner and extended by Leader is reproved from the point of view of “differentiation”. As a by-product, it is shown that every non-atomic probability measure is in the closed convex hull (topology of simple convergence) of those zero-one-valued additive set functions which are not countably additive; a curious result when applied to Lebesgue measure.

Mathematical Subject Classification 2000
Primary: 28B15
Received: 9 January 1980
Revised: 24 September 1980
Published: 1 February 1983
P. H. Maserick