Block averages as conditional expectations

The block-average step graphon of a measurable finite partition agrees almost everywhere with conditional expectation onto the σ-algebra recording the partition part of each coordinate. This identifies the strict block-average construction with the analytic conditional-expectation API, so approximation along refining partitions can use martingale convergence.

The identification includes partitions with null parts: the strict representative uses zero on null rectangles, while conditional expectation determines values only almost everywhere.

References

• L. Lovász, Large Networks and Graph Limits, AMS Colloquium Publications 60 (2012), §9.2.

theorem TauCeti.DenseGraphLimits.stepGraphonAvg_ae_eq_condExp {Ω : Type u_1} [MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (P : Finpartition Set.univ) (hP : ∀ p ∈ P.parts, MeasurableSet p) (W : Graphon Ω μ) : (fun (z : Ω × Ω) => (stepGraphonAvg P hP W) z.1 z.2) =ᵐ[μ.prod μ] (μ.prod μ)[fun (z : Ω × Ω) => W z.1 z.2 | MeasurableSpace.comap (Prod.map P.indexedPartition.index P.indexedPartition.index) ⊤]

Block averaging is conditional expectation given the two partition indices. The equality is almost everywhere, so it is independent of values on null rectangles.