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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 | import OAI.Probability.InvariantIsing.Core.FiniteIndexCoupling
import OAI.Probability.InvariantIsing.Spectral.MeasureMagneticTransport
namespace OAI
/-! The magnetic transportation estimate on a common probability space. -/
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped BigOperators
namespace InvariantIsing
lemma finiteIndex_magnetic_sub_le {Ω A B : Type*} [MeasurableSpace Ω]
[Fintype A] [MeasurableSpace A] [MeasurableSingletonClass A]
[Fintype B] [MeasurableSpace B] [MeasurableSingletonClass B]
(P : Measure Ω) [IsProbabilityMeasure P]
(g : Ω → A) (h : Ω → B) (hg : Measurable g) (hh : Measurable h)
(c : A → ℝ) (d : B → ℝ) (hpos : ∀ b, 0 < finiteIndexWeight P h b)
(ν : ProbabilityMeasure ℝ) (a b : ℝ)
(hcompact : IsCompact (ν : Measure ℝ).support)
(hbound : (ν : Measure ℝ).support ⊆ Icc a b)
(ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support) :
(finiteMagneticFunctional (measureR (ν : Measure ℝ) b) (finiteIndexWeight P g) c).toReal-
(finiteMagneticFunctional (measureR (ν : Measure ℝ) b) (finiteIndexWeight P h) d).toReal ≤
∫ x, |c (g x)-d (h x)| ∂P := by
have ht := measureMagnetic_transport_le ν a b hcompact hbound ha hb
(finiteIndexCoupling P g h) (finiteIndexWeight P g) c (finiteIndexWeight P h) d
(fun _ _ => measureReal_nonneg) (fun _ => measureReal_nonneg) hpos
(finiteIndexWeight_sum P g hg) (finiteIndexWeight_sum P h hh)
(finiteIndexCoupling_row P g h hh) (finiteIndexCoupling_col P g h hg)
rw [← finiteIndexCoupling_integral P g h hg hh] at ht
linarith
end InvariantIsing
end
end OAI
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