A note on the "conditional triviality" property for regular conditional distributions

Today, we dive into a deep observation concerning regular conditional distributions (rcd), a key concept in the measure-theoretic versions of the ergodic decomposition theorem. We will unravel the relation between regular conditional distributions and the "conditional triviality" property.

Background and Definitions

Assume (X,Σ) as a countably generated measurable space, and P is the set of probability measures on (X,Σ), equipped with its naturalσ-algebra. A regular conditional distribution (rcd) of ρ given G is a (G,K)-measurable function ρG: X→P such that for all A∈Σ and G∈ G, ρ(A∩G) = ∫ G ρG(x)(A)ρ(dx).

If (X,Σ) is countably generated, a function ρG: X→P exists and it is ρSG-essentially unique. If (X,Σ) is standard, such a function ρG must exist (as per Theorem 33.3) and going forward, ρG will denote a rcd of ρ given G.

Remarks and Lemma

Two conditions are equivalent: (i) G is ρ-trivial; (ii) the constant map x↦ρ is a rcd of ρ given G; and (iii) ρG : X→P is ρ-almost constant. A function ρG2 : X×X→P will be called an iterated rcd of ρ given G if for almost every x∈X, the map y↦ρG2(x,y) is a rcd of ρG(x) given G.

Theorem and Proof

The theorem states that G is conditionally trivial under ρ if and only if there exists an iterated rcd of ρ given G which is (G ⊗ G,K)-measurable. The proof can be derived using the correlation between the conditional triviality and the measurable iterated rcd. For instance, if G is conditionally trivial under ρ then the map (x,y)↦ρG(x) is an iterated rcd of ρ given G, and it is (G ⊗ G,K)-measurable.

Final Notes

We note that although conditional triviality does not always hold, it serves to demonstrate the non-measurability or non-constructivity of the operation of conditioning with respect to a σ-algebra. This fact becomes more evident when G has the property that there exists a (K⊗ G,K)-measurable function C_G : P ×X→P such that for all μ∈ P the map x↦C_G(μ,x) is a rcd of μ given G. Therefore, the map (x,y)↦C_G(C_G(ρ,x),y) is a (G ⊗ G,K)-measurable iterated rcd of ρ given G, implying that G is conditionally trivial under ρ.

Acknowledgement

The author would like to extend thanks to Prof. Patrizia Berti and Prof. Pietro Rigo for their helpful comments and kind encouragement.

References

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