File size: 1,302 Bytes
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 | import OAI.Probability.InvariantIsing.Spectral.ResolventPowerDerivative
namespace OAI
/-! Equality of positive resolvents determines all their bounded powers. -/
noncomputable section
open MeasureTheory ProbabilityTheory Set Filter
open scoped Topology
namespace InvariantIsing
theorem positiveResolvent_moments_eq (μ ν : Measure ℝ) [IsFiniteMeasure μ] [IsFiniteMeasure ν]
(h : ∀ t : ℝ, 0 < t → (∫ x, positiveResolventTest t x ∂μ) =
∫ x, positiveResolventTest t x ∂ν) :
∀ n : ℕ, ∀ t : ℝ, 0 < t →
(∫ x, (positiveResolventTest t x)^(n+1) ∂μ) =
∫ x, (positiveResolventTest t x)^(n+1) ∂ν := by
intro n
induction n with
| zero => simpa only [zero_add,pow_one] using h
| succ n ih =>
intro t ht
have he : (fun s => ∫ x, (positiveResolventTest s x)^(n+1) ∂ν) =ᶠ[𝓝 t]
fun s => ∫ x, (positiveResolventTest s x)^(n+1) ∂μ := by
filter_upwards [Ioi_mem_nhds ht] with s hs
exact (ih s hs).symm
have hd := ((positiveResolventPower_integral_hasDerivAt μ ht (n+1)).congr_of_eventuallyEq he).unique
(positiveResolventPower_integral_hasDerivAt ν ht (n+1))
exact mul_left_cancel₀ (neg_ne_zero.mpr (Nat.cast_ne_zero.mpr (Nat.succ_ne_zero n))) hd
end InvariantIsing
end
end OAI
|