| import OAI.Probability.InvariantIsing.Core.FiniteIndexDistribution |
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| namespace OAI |
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| /-! Joint finite random labels give exact nonnegative transport plans. -/ |
| noncomputable section |
| open MeasureTheory Set |
| open scoped BigOperators Classical |
| namespace InvariantIsing |
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| variable {Ω A B : Type*} [MeasurableSpace Ω] [Fintype A] [Fintype B] |
| [MeasurableSpace A] [MeasurableSingletonClass A] |
| [MeasurableSpace B] [MeasurableSingletonClass B] |
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| def finiteIndexCoupling (P : Measure Ω) (g : Ω → A) (h : Ω → B) (a : A) (b : B) : ℝ := |
| P.real ((g ⁻¹' {a}) ∩ (h ⁻¹' {b})) |
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| omit [Fintype A] [MeasurableSpace A] [MeasurableSingletonClass A] in |
| lemma finiteIndexCoupling_row (P : Measure Ω) [IsProbabilityMeasure P] |
| (g : Ω → A) (h : Ω → B) (hh : Measurable h) (a : A) : |
| ∑ b, finiteIndexCoupling P g h a b=finiteIndexWeight P g a := by |
| have hs := sum_measureReal_preimage_singleton (μ := P.restrict (g ⁻¹' {a})) |
| Finset.univ (f := h) (fun b _ => hh (measurableSet_singleton b)) |
| simp only [measureReal_restrict_apply (hh (measurableSet_singleton _)), |
| Finset.coe_univ,preimage_univ,measureReal_restrict_apply_univ] at hs |
| simpa only [finiteIndexCoupling,finiteIndexWeight,inter_comm] using hs |
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| omit [Fintype B] [MeasurableSpace B] [MeasurableSingletonClass B] in |
| lemma finiteIndexCoupling_col (P : Measure Ω) [IsProbabilityMeasure P] |
| (g : Ω → A) (h : Ω → B) (hg : Measurable g) (b : B) : |
| ∑ a, finiteIndexCoupling P g h a b=finiteIndexWeight P h b := by |
| simpa only [finiteIndexCoupling,inter_comm] using finiteIndexCoupling_row P h g hg b |
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| lemma finiteIndexCoupling_integral (P : Measure Ω) [IsProbabilityMeasure P] |
| (g : Ω → A) (h : Ω → B) (hg : Measurable g) (hh : Measurable h) |
| (f : A → B → ℝ) : |
| (∫ x, f (g x) (h x) ∂P)=∑ a, ∑ b, finiteIndexCoupling P g h a b*f a b := by |
| let T := fun a b => (g ⁻¹' {a}) ∩ (h ⁻¹' {b}) |
| have hT a b : MeasurableSet (T a b) := |
| (hg (measurableSet_singleton a)).inter (hh (measurableSet_singleton b)) |
| have hf : (fun x => f (g x) (h x))= |
| fun x => ∑ a, ∑ b, (T a b).indicator (fun _ => f a b) x := by |
| funext x |
| simp [T,Set.indicator_apply,ite_and] |
| rw [hf,integral_finsetSum] |
| · apply Finset.sum_congr rfl |
| intro a _ |
| rw [integral_finsetSum] |
| · apply Finset.sum_congr rfl |
| intro b _ |
| rw [integral_indicator_const _ (hT a b)] |
| rfl |
| · intro b _ |
| exact (integrable_const _).indicator (hT a b) |
| · intro a _ |
| exact integrable_finsetSum _ (fun b _ => (integrable_const _).indicator (hT a b)) |
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| end InvariantIsing |
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| end |
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| end OAI |
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