| import OAI.Geometry.PinnedDistances.FiniteLaw |
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| namespace OAI |
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| universe u1 u2 u3 u4 u5 |
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| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped Topology |
| namespace WeakPinned.Bounded |
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| lemma integrable_one_bound {X : Type u1} [MeasurableSpace X] (μ : Measure X) [IsFiniteMeasure μ] |
| (f : X → ℝ) (hf : Measurable f) (hbound : ∀ x,‖f x‖≤1) : Integrable f μ := by |
| exact (integrable_const (1:ℝ)).mono' hf.aestronglyMeasurable (Eventually.of_forall hbound) |
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| variable {Ω : Type u2} {E : Type u3} {X : Type u4} {Y : Type u5} [MeasurableSpace Ω] [MeasurableSpace E] |
| [MeasurableSpace X] [MeasurableSpace Y] |
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| lemma integral_fiber_indicator (Q : Measure E) (μ : Measure X) |
| [IsProbabilityMeasure Q] [IsProbabilityMeasure μ] |
| (f : E × X → ℝ) (hf : Measurable f) (hbound : ∀ p,‖f p‖≤1) |
| (s : Set E) (hs : MeasurableSet s) : |
| (∫ e in s, ∫ x,f (e,x) ∂ μ ∂ Q)= |
| ∫ p,(s ×ˢ Set.univ).indicator f p ∂ Q.prod μ := by |
| have hi := integrable_one_bound (Q.prod μ) f hf hbound |
| rw [integral_prod _ (hi.indicator (hs.prod MeasurableSet.univ))] |
| rw [← integral_indicator hs] |
| apply integral_congr_ae |
| filter_upwards [] with e |
| by_cases he : e∈s |
| · simp [Set.indicator_of_mem,he] |
| · simp [Set.indicator_of_notMem,he] |
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| lemma conditional_integral_eq |
| (R : Measure Ω) (Q : Measure E) (μ : Measure X) (ν : Measure Y) |
| [IsProbabilityMeasure R] [IsProbabilityMeasure Q] |
| [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| (e : Ω → E) (z : Ω → X) (w : Ω → Y) |
| (he : Measurable e) (hz : Measurable z) (hw : Measurable w) |
| (hmz : R.map (fun r => (e r,z r))=Q.prod μ) |
| (hmw : R.map (fun r => (e r,w r))=Q.prod ν) |
| (f : E × X → ℝ) (g : E × Y → ℝ) |
| (hf : Measurable f) (hg : Measurable g) |
| (hfb : ∀ p,‖f p‖≤1) (hgb : ∀ p,‖g p‖≤1) |
| (heq : ∀ᵐ r ∂ R,f (e r,z r)=g (e r,w r)) : |
| ∀ᵐ a ∂ Q,(∫ x,f (a,x) ∂ μ)=(∫ y,g (a,y) ∂ ν) := by |
| apply Integrable.ae_eq_of_forall_setIntegral_eq |
| · exact (integrable_one_bound (Q.prod μ) f hf hfb).integral_prod_left |
| · exact (integrable_one_bound (Q.prod ν) g hg hgb).integral_prod_left |
| · intro s hs _ |
| rw [integral_fiber_indicator Q μ f hf hfb s hs, |
| integral_fiber_indicator Q ν g hg hgb s hs,← hmz,← hmw] |
| rw [integral_map_of_stronglyMeasurable (he.prodMk hz) |
| (hf.indicator (hs.prod MeasurableSet.univ)).stronglyMeasurable, |
| integral_map_of_stronglyMeasurable (he.prodMk hw) |
| (hg.indicator (hs.prod MeasurableSet.univ)).stronglyMeasurable] |
| apply integral_congr_ae |
| filter_upwards [heq] with r hr |
| by_cases h : e r∈s <;> simp [h,hr] |
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| end WeakPinned.Bounded |
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| end |
| end |
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| end OAI |
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