Lifting theory

Existence of liftings

Theorem. Suppose (X, Σ, μ) is complete. Then (X, Σ, μ) admits a lifting if and only if there exists a collection of mutually disjoint integrable sets in Σ whose union is X. In particular, if (X, Σ, μ) is the completion of a σ-finite measure or of an inner regular Borel measure on a locally compact space, then (X, Σ, μ) admits a lifting.

The proof consists in extending a lifting to ever larger sub-σ-algebras, applying Doob's martingale convergence theorem if one encounters a countable chain in the process.

Strong liftings

Suppose (X, Σ, μ) is complete and X is equipped with a completely regular Hausdorff topology τ ⊂ Σ such that the union of any collection of negligible open sets is again negligible – this is the case if (X, Σ, μ) is σ-finite or comes from a Radon measure. Then the support of μ, Supp(μ), can be defined as the complement of the largest negligible open subset, and the collection C_b(X, τ) of bounded continuous functions belongs to L ∞ ( X , Σ , μ ) .

A strong lifting for (X, Σ, μ) is a lifting

such that Tφ = φ on Supp(μ) for all φ in C_b(X, τ). This is the same as requiring that TU ≥ (U ∩ Supp(μ)) for all open sets U in τ.

Theorem. If (Σ, μ) is σ-finite and complete and τ has a countable basis then (X, Σ, μ) admits a strong lifting.

Proof. Let T₀ be a lifting for (X, Σ, μ) and {U₁, U₂, ...} a countable basis for τ. For any point p in the negligible set

let T_p be any character on L^∞(X, Σ, μ) that extends the character φ ↦ φ(p) of C_b(X, τ). Then for p in X and [f] in L^∞(X, Σ, μ) define:

T is the desired strong lifting.

Application: disintegration of a measure

Suppose (X, Σ, μ), (Y, Φ, ν) are σ-finite measure spaces (μ, ν positive) and π : X → Y is a measurable map. A disintegration of μ along π with respect to ν is a slew Y ∋ y ↦ λ y of positive σ-additive measures on (X, Σ) such that

1. λ_y is carried by the fiber π − 1 ( { y } ) of π over y:

1. for every μ-integrable function f,

Disintegrations exist in various circumstances, the proofs varying but almost all using strong liftings. Here is a rather general result. Its short proof gives the general flavor.

Theorem. Suppose X is a Polish space and Y a separable Hausdorff space, both equipped with their Borel σ-algebras. Let μ be a σ-finite Borel measure on X and π : X → Y a Σ, Φ–measurable map. Then there exists a σ-finite Borel measure ν on Y and a disintegration (*). If μ is finite, ν can be taken to be the pushforward π_∗μ, and then the λ_y are probabilities.

Proof. Because of the polish nature of X there is a sequence of compact subsets of X that are mutually disjoint, whose union has negligible complement, and on which π is continuous. This observation reduces the problem to the case that both X and Y are compact and π is continuous, and ν = π_∗μ. Complete Φ under ν and fix a strong lifting T for (Y, Φ, ν). Given a bounded μ-measurable function f, let ⌊ f ⌋ denote its conditional expectation under π, i.e., the Radon-Nikodym derivative of π_∗(fμ) with respect to π_∗μ. Then set, for every y in Y, λ y ( f ) := T ( ⌊ f ⌋ ) ( y ) . To show that this defines a disintegration is a matter of bookkeeping and a suitable Fubini theorem. To see how the strongness of the lifting enters, note that

and take the infimum over all positive φ in C_b(Y) with φ(y) = 1; it becomes apparent that the support of λ_y lies in the fiber over y.