buying_to_bundle_quality_mul_allocation_integrable

Proved

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

asymptoticseconomicsmechanism-designprobability

Under the market support assumptions and 0 <= x <= 1, the mean-quality payment integrand m * x m is integrable with respect to the quality distribution.

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.Integral.Bochner.Basic
import Definitions.Def_buying_to_bundle_market

open MeasureTheory

Formal statement

theorem buying_to_bundle_quality_mul_allocation_integrable
    (σ μL μH γ ξ : ℝ) (qual noise : Measure ℝ)
    [IsProbabilityMeasure qual] [IsProbabilityMeasure noise]
    (qualPdf noisePdf : ℝ → ℝ)
    (hM : BuyingToBundle.MarketAssumptions σ μL μH γ ξ qual noise qualPdf noisePdf)
    (x : ℝ → ℝ) (hx : BuyingToBundle.IsAllocationRule μL μH x) :
    Integrable (fun m => m * x m) qual := by sorry

Source

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

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