| import Mathlib | |
| import OAI.Analysis.HardSphere.Geometry.KilledObservable | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped ENNReal Topology BigOperators | |
| open scoped BigOperators ComplexConjugate | |
| open MeasureTheory ProbabilityTheory Filter Set | |
| open scoped ENNReal NNReal Topology BigOperators | |
| open MeasureTheory ProbabilityTheory Real Complex | |
| open scoped ENNReal ComplexConjugate | |
| namespace HardSphere | |
| lemma killedTransition_continuous_ae (N : ℕ) (L : Length) (a t : ℝ) | |
| (g : C(Configuration N L,ℂ)) : | |
| killedTransition N L a t (ContinuousMap.toLp 2 volume ℂ g) =ᵐ[volume] | |
| killedAverage N L a t g := by | |
| have hgLp : MemLp (fun X ↦ g X) 2 (volume : Measure (Configuration N L)) := | |
| MemLp.of_bound g.continuous.measurable.aestronglyMeasurable ‖g‖ | |
| (Eventually.of_forall g.norm_coe_le_norm) | |
| have hge : ContinuousMap.toLp 2 volume ℂ g = hgLp.toLp g := by | |
| apply Lp.ext | |
| exact (ContinuousMap.coeFn_toLp (𝕜 := ℂ) (p := 2) volume g).trans hgLp.coeFn_toLp.symm | |
| rw [hge,killedTransition_bounded N L a t g.continuous.measurable | |
| hgLp (norm_nonneg g) g.norm_coe_le_norm] | |
| exact MemLp.coeFn_toLp _ | |
| end HardSphere | |
| end | |
| end OAI | |