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a89d282 | 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 | import OAI.Probability.InvariantIsing.Core.UnitIntervalMomentUniqueness
namespace OAI
/-! Positive resolvents determine probability measures on the nonnegative real line. -/
noncomputable section
open MeasureTheory ProbabilityTheory Set
namespace InvariantIsing
def resolventUnit (x : ℝ) : Icc (0 : ℝ) 1 :=
⟨positiveResolventTest 1 x, (positiveResolventTest_bound (t := 1) (by norm_num) x).1,
by simpa only [one_div_one] using (positiveResolventTest_bound (t := 1) (by norm_num) x).2⟩
lemma continuous_resolventUnit : Continuous resolventUnit :=
(continuous_positiveResolventTest (by norm_num : (0 : ℝ) < 1)).subtype_mk _
lemma resolventUnit_inverse {x : ℝ} (hx : 0 ≤ x) : 1/(resolventUnit x : ℝ)-1 = x := by
change 1/(1/(1+max x 0))-1=x
rw [max_eq_left hx,one_div_one_div]
ring
lemma resolventUnit_map_inverse (μ : Measure ℝ) (hμ : ∀ᵐ x ∂μ, 0 ≤ x) :
(μ.map resolventUnit).map (fun x : Icc (0 : ℝ) 1 => 1/(x : ℝ)-1) = μ := by
have hm : Measurable (fun x : Icc (0 : ℝ) 1 => 1/(x : ℝ)-1) := by fun_prop
rw [Measure.map_map hm continuous_resolventUnit.measurable]
calc
μ.map ((fun x : Icc (0 : ℝ) 1 => 1/(x : ℝ)-1) ∘ resolventUnit) = μ.map id := by
apply Measure.map_congr
filter_upwards [hμ] with x hx
exact resolventUnit_inverse hx
_ = μ := Measure.map_id
theorem positiveResolvent_unique (μ ν : Measure ℝ) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
(hμ : ∀ᵐ x ∂μ, 0 ≤ x) (hν : ∀ᵐ x ∂ν, 0 ≤ x)
(h : ∀ t : ℝ, 0 < t → (∫ x, positiveResolventTest t x ∂μ) =
∫ x, positiveResolventTest t x ∂ν) : μ = ν := by
have : IsProbabilityMeasure (μ.map resolventUnit) :=
(Measure.isProbabilityMeasure_map_iff continuous_resolventUnit.measurable.aemeasurable).mpr inferInstance
have : IsProbabilityMeasure (ν.map resolventUnit) :=
(Measure.isProbabilityMeasure_map_iff continuous_resolventUnit.measurable.aemeasurable).mpr inferInstance
have he : μ.map resolventUnit = ν.map resolventUnit := by
apply unitInterval_moments_unique
intro n
have hc : Continuous (fun x : Icc (0 : ℝ) 1 => (x : ℝ)^n) := continuous_subtype_val.pow n
rw [integral_map continuous_resolventUnit.measurable.aemeasurable hc.aestronglyMeasurable,
integral_map continuous_resolventUnit.measurable.aemeasurable hc.aestronglyMeasurable]
cases n with
| zero => simp only [pow_zero,integral_const,probReal_univ,smul_eq_mul,mul_one]
| succ n => exact positiveResolvent_moments_eq μ ν h n 1 (by norm_num)
have hi := congrArg (fun η : Measure (Icc (0 : ℝ) 1) =>
η.map (fun x : Icc (0 : ℝ) 1 => 1/(x : ℝ)-1)) he
rwa [resolventUnit_map_inverse μ hμ,resolventUnit_map_inverse ν hν] at hi
end InvariantIsing
end
end OAI
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