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) =∫Γρ(x)dμf(ρ) where either
dμf= f′dμ with 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
Vp-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.
