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It is shown that Gelfand transforms of elements f∈L ∞ (μ) \udf∈L∞(μ)\ud are almost constant at almost every fiber Π −1 ({x}) \udΠ−1({x})\ud of the spectrum of L ∞(μ) in the following sense: for each f∈L ∞ (μ) \udf∈L∞(μ)\ud 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 \udΠ−1({x})∩U\ud. As an application a new approach to disintegration of measures is presented, allowing one to drop the usually taken separability assumption

Fibers of the L ∞ algebra and disintegration of measures