Fibers of the L~∞ algebra and disintegration of measures

摘要

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.