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9eeb9c7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 | import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.Probability.Notation
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped ProbabilityTheory
namespace Invariant233.ConditionalEventLocalization
variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
{μ : Measure Ω} [IsFiniteMeasure μ] {m : MeasurableSpace Ω}
theorem condProb_mono_on (hm : m ≤ mΩ) {A B S : Set Ω}
(hA : MeasurableSet[mΩ] A) (hB : MeasurableSet[mΩ] B) (hS : MeasurableSet[m] S)
(hsub : ∀ᵐ ω ∂μ, ω ∈ S → ω ∈ A → ω ∈ B) :
∀ᵐ ω ∂μ, ω ∈ S → (μ⟦A | m⟧) ω ≤ (μ⟦B | m⟧) ω := by
have hAi : Integrable (A.indicator (fun _ => (1 : ℝ))) μ :=
(integrable_const _).indicator hA
have hBi : Integrable (B.indicator (fun _ => (1 : ℝ))) μ :=
(integrable_const _).indicator hB
have hle : S.indicator (A.indicator (fun _ => (1 : ℝ))) ≤ᵐ[μ]
S.indicator (B.indicator (fun _ => (1 : ℝ))) := by
filter_upwards [hsub] with ω hω
by_cases hs : ω ∈ S
· simp only [indicator_of_mem hs]
by_cases ha : ω ∈ A
· simp only [indicator_of_mem ha, indicator_of_mem (hω hs ha), le_refl]
· simp only [indicator_of_notMem ha]
exact indicator_nonneg (fun _ _ => by norm_num) _
· simp only [indicator_of_notMem hs, le_refl]
have hc := condExp_mono (m := m) (hAi.indicator (hm _ hS))
(hBi.indicator (hm _ hS)) hle
filter_upwards [hc, condExp_indicator hAi hS, condExp_indicator hBi hS]
with ω hle hleft hright
intro hs
rw [hleft, hright, indicator_of_mem hs, indicator_of_mem hs] at hle
exact hle
theorem condProb_lower_on_of_forcing (hm : m ≤ mΩ) {A B S : Set Ω}
(hA : MeasurableSet[mΩ] A) (hB : MeasurableSet[mΩ] B) (hS : MeasurableSet[m] S)
(hsub : ∀ᵐ ω ∂μ, ω ∈ S → ω ∈ A → ω ∈ B)
(r : Ω → ℝ) (hlower : ∀ᵐ ω ∂μ, ω ∈ S → r ω ≤ (μ⟦A | m⟧) ω) :
∀ᵐ ω ∂μ, ω ∈ S → r ω ≤ (μ⟦B | m⟧) ω := by
filter_upwards [@condProb_mono_on Ω mΩ μ _ m hm A B S hA hB hS hsub, hlower] with ω hmono hlo
exact fun hs => (hlo hs).trans (hmono hs)
end Invariant233.ConditionalEventLocalization
end
end OAI
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