File size: 1,201 Bytes
9eeb9c7 | 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 | 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
|