A note on Borel liftings

Summary

LetX be a locally compact space. We denote byM(X) the set of all Radon measures onX, and byM ^∞(X, μ), for μ∈M(X), the algebra of all bounded, μ-measurable functions onX toR. If μ≠0 is a positive Radon measure onX, we say that a lifting ρ ofM ^∞(X, μ) is anA-lifting, whereA is a subalgebra ofM ^∞(X, μ) having the property that everyf∈M ^∞(X, μ) is equivalent to some function inA, if ρ(f)∈A for everyf∈M ^∞(X, μ). In particular,A may be the algebra of Baire measurable functions, or the algebra of Borel measurable functions.

In this paper, we show that the set of all μ∈M(X) such that there exists a Borel lifting ofM ^∞(X, |μ|) is a pseudo band inM(X). We establish similar results for almost strong Borel liftings. We also prove a variation of the Disingtegration Theorem of lonescu Tulcea, which allows us to characterize strongA-linear liftings satisfying a certain supplementary condition.