buying_to_bundle_intermediate_surrogate_mean_quality_product_integrable

Proved

by ann · Jul 3, 2026 · Mathlib c5ea003 (Lean v4.30.0)

asymptoticseconomicsmechanism-designprobability

Integrability over qual^N of the deterministic mean-quality term sum_i x(mu_i) mu_i, derived from one-dimensional integrability.

Source: Buying to Bundle: Optimal Sourcing from Monopolistic Sellers, Appendix C.2, proof of Theorem 4.6, pp. 35-36 (and Lemma 4.5, pp. 33-35, where applicable).

Preamble

import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Definitions.Def_buying_to_bundle_market

open MeasureTheory

Formal statement

theorem buying_to_bundle_intermediate_surrogate_mean_quality_product_integrable
    (N : ℕ) (qual : Measure ℝ) [IsProbabilityMeasure qual]
    (x : ℝ → ℝ)
    (hμx : Integrable (fun m => m * x m) qual) :
    Integrable (fun μ : Fin N → ℝ => ∑ i, x (μ i) * μ i)
      (Measure.pi fun _ : Fin N => qual) := by sorry

Source

Buying to Bundle: Optimal Sourcing from Monopolistic Sellers, Appendix C.2, proof of Theorem 4.6

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