buying_to_bundle_intermediate_surrogate_revenue_integrable

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

asymptoticseconomicsmechanism-designprobability

Integrability of the intermediate-surrogate monopoly-revenue integrand over the product 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.Constructions.Pi
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Definitions.Def_buying_to_bundle_market

open MeasureTheory

Formal statement

theorem buying_to_bundle_intermediate_surrogate_revenue_integrable
    (N : ℕ) (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 μ : Fin N → ℝ =>
        BuyingToBundle.monopolyRevenue
          ((Measure.pi fun _ : Fin N => noise).map
            fun z => ∑ i, x (μ i) * (μ i + σ * z 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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