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7f37411 | 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 | import OAI.MeasureTheory.Falconer.Shells.HighShellOrderForm
namespace OAI
namespace PlanarFalconer
open MeasureTheory
namespace PositiveDistanceData
theorem normalizedHighShellF_measurable (d : PositiveDistanceData) [IsFiniteMeasure d.ρ]
(hQ : 0 < d.Q.side) (hs : d.StronglySeparated) (n : ℕ) (v : ℝ) :
Measurable (d.normalizedHighShellF n v) := by
have he (r : ℝ) : d.normalizedHighShellF n v r =
(((r*(2:ℝ)^(-(d.a:ℝ)))^(1/2:ℝ):ℝ):ℂ) *
∑ j ∈ Finset.range (n+1), ((((r/(2:ℝ)^v)⁻¹)^j:ℝ):ℂ)*d.inverseOrderForm j v r :=
d.normalizedHighShellF_finite_separation hQ hs n v r
simp_rw [show d.normalizedHighShellF n v = _ from funext he]
apply Measurable.mul
· fun_prop
· apply Finset.measurable_sum
intro j hj
exact (by fun_prop : Measurable (fun r : ℝ => ((((r/(2:ℝ)^v)⁻¹)^j:ℝ):ℂ))).mul
(d.inverseOrderForm_data hQ j v).1
end PositiveDistanceData
end PlanarFalconer
end OAI
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