Fibers of the L∞ algebra and disintegration of measures

Abstract

It is shown that Gelfand transforms of elements f ∈ L∞(μ) are almost constant at almost every fiber Π-1({x}) of the spectrum of L∞(μ) in the following sense: for each f ∈ L∞(μ) there is an open dense subset U = U(f) of this spectrum having full measure and such that the Gelfand transform of f is constant on the intersection Π-1({x}) ∩ U. As an application a new approach to disintegration of measures is presented, allowing one to drop the usually taken separability assumption. © 2011 The Author(s).