| 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 | |