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c4cdbbd | 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 55 56 57 58 59 60 61 62 63 | import OAI.Probability.InvariantIsing.Core.FiniteIndexDistribution
namespace OAI
/-! Joint finite random labels give exact nonnegative transport plans. -/
noncomputable section
open MeasureTheory Set
open scoped BigOperators Classical
namespace InvariantIsing
variable {Ω A B : Type*} [MeasurableSpace Ω] [Fintype A] [Fintype B]
[MeasurableSpace A] [MeasurableSingletonClass A]
[MeasurableSpace B] [MeasurableSingletonClass B]
def finiteIndexCoupling (P : Measure Ω) (g : Ω → A) (h : Ω → B) (a : A) (b : B) : ℝ :=
P.real ((g ⁻¹' {a}) ∩ (h ⁻¹' {b}))
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
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
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))
end InvariantIsing
end
end OAI
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