Strong-Markov lower integrals of stopped-history restricted future path laws

import Mathlib.MeasureTheory.Integral.Lebesgue.Map
import Mathlib.Probability.Process.Stopping
import Definitions.Def_AvramDividend_Classical_SpectrallyNegativeLevy
import Theorems.Thm_AvramDividend_Classical_stoppedHistory_event_measurable
import Theorems.Thm_AvramDividend_Classical_levy_finiteStopping_stoppedFuturePath_measurable
import Theorems.Thm_AvramDividend_Classical_levy_finiteStopping_scalar_markov_no_meas_premise

open MeasureTheory ProbabilityTheory Filter Set
open scoped NNReal ENNReal
open AvramDividend.Classical

theorem AvramDividend.Classical.levy_finiteStopping_restrictedPath_continuousReward_lintegral
    {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω}
    {𝓕 : Filtration ℝ≥0 mΩ}
    (X : SpectrallyNegativeLevy P 𝓕)
    (τ : Ω → ℝ≥0)
    (hstop : IsStoppingTime 𝓕 (fun ω => (τ ω : WithTop ℝ≥0)))
    (A : Set Ω)
    (hA : ∀ s : ℝ≥0,
      MeasurableSet[𝓕 s] (A ∩ {ω : Ω | τ ω ≤ s}))
    (g : (ℝ≥0 → ℝ) → ℝ≥0∞)
    (hg : Continuous g) (hgm : Measurable g)
    (C : ℝ≥0∞) (hbound : ∀ y, g y ≤ C)
    (hfin : (∫⁻ _ : Ω, C ∂P) ≠ ∞) :
    (∫⁻ y, g y ∂(Measure.map
      (fun ω : Ω => fun t : ℝ≥0 =>
        X.X (τ ω + t) ω - X.X (τ ω) ω) (P.restrict A))) =
    P A * (∫⁻ y, g y ∂(Measure.map
      (fun ω : Ω => fun t : ℝ≥0 => X.X t ω) P)) := by
  sorry