Every graphon is pulled back from a standard Borel carrier

A graphon is jointly measurable, so it depends on only countably many measurable sets of each argument: it factors as W x y = V (q x) (q y) through a measurable q : Ω → ℕ → Bool into the Cantor space, and the factor V is again symmetric and [0, 1]-valued (TauCeti.DenseGraphLimits.Graphon.exists_comap_natBool). Since q is measure preserving onto the pushforward μ.map q, this presents an arbitrary graphon as a pullback W = V.comap q of a graphon on a standard Borel probability carrier.

This is the carrier-free half of Janson's Lemma 7.3, and it is what removes the standard Borel hypothesis from the arguments that need one: the measure-theoretic input has no hypothesis on Ω at all, because the factorization happens before any measure is involved.

Symmetry and the range constraint are retained, not re-imposed: the factor produced by the measurable factorization need be neither, so it is averaged with its transpose and truncated to [0, 1]. Both operations are invisible on the image of q, where the values are already symmetric and in [0, 1].

Main results

• TauCeti.DenseGraphLimits.Graphon.exists_comap_natBool — every graphon is the pullback, along a measurable map to the Cantor space, of a graphon on the pushforward measure.

References

• S. Janson, Graphons, cut norm and distance, couplings and rearrangements, NYJM Monographs 4 (2013), Lemma 7.3.

theorem TauCeti.DenseGraphLimits.Graphon.exists_comap_natBool {Ω : Type u_1} [MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (W : Graphon Ω μ) : ∃ (q : Ω → ℕ → Bool) (hq : Measurable q) (V : Graphon (ℕ → Bool) (MeasureTheory.Measure.map q μ)), V.comap q hq μ = W

Every graphon is a pullback from a standard Borel carrier (Janson, Lemma 7.3). There is a measurable q : Ω → ℕ → Bool and a graphon V on the Cantor space, carrying the pushforward measure μ.map q, with W = V.comap q; no hypothesis on Ω is needed.

The map q is measure preserving from μ onto μ.map q by construction, so W and V have the same observables; this is the reduction that lets a statement proved over standard Borel carriers be transported to an arbitrary probability carrier.