Applications of differentiation of ℒp-functions to semilattices
Peter H. Maserick · Pacific Journal of Mathematics · 1983
Let S be a commutative semigroup with identity 1 such that x 2 = x for each x E 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/on S which admit disintegrations of the form f(x) = f τ p( x) dμy(ρ) where either dμ,f = f'dμ with/' E L p (μ) (1 < p < oo) or μ f 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 oo to all p E [1, oo].Applications of this theory to the classical L^-spaces on the w-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 F^-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.