| import Mathlib |
| import OAI.Analysis.BrownMeasure.Atomic |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter InnerProductSpace |
| open scoped ComplexInnerProductSpace Topology |
| namespace PolarStructure |
| variable {Ω ι : Type*} [MeasurableSpace Ω] [Countable ι] {μ : Measure Ω} |
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|
| theorem jointPurePoint_of_countable_levels |
| (A : ι → (Lp ℂ 2 μ →L[ℂ] Lp ℂ 2 μ)) (r : Ω → ι → ℝ) |
| (hr : Measurable r) |
| (hA : ∀ i u,A i u =ᵐ[μ] fun x=>(r x i:ℂ)*u x) |
| (S : Set (ι → ℝ)) (hS : S.Countable) (hmem : ∀ᵐ x ∂μ,r x∈S) : JointPurePoint A := by |
| let K := (Submodule.span ℂ {x | ∃ t : ι → ℝ,x∈jointSpace A t}).topologicalClosure |
| have hK : Kᗮ=⊥ := by |
| apply (Submodule.eq_bot_iff _).mpr |
| intro u hu |
| have hz (t : ι → ℝ) : ∀ᵐ x ∂μ,r x=t → u x=0 := by |
| let E := r ⁻¹' {t} |
| have hE : MeasurableSet E := hr (measurableSet_singleton t) |
| let hv := (Lp.memLp u).indicator hE |
| let v : Lp ℂ 2 μ := hv.toLp (E.indicator u) |
| have hvE : v =ᵐ[μ] E.indicator u := hv.coeFn_toLp |
| have hvj : v∈jointSpace A t := by |
| intro i |
| apply Lp.ext |
| filter_upwards [hA i v,hvE,Lp.coeFn_smul (t i:ℂ) v] with x hx hxe hxs |
| rw [hx,hxs] |
| by_cases hxE : x∈E |
| · have he : r x=t := hxE |
| simp [he] |
| · have h0 : v x=0 := by rw [hxe,Set.indicator_of_notMem hxE] |
| simp [h0] |
| have hvK : v∈K := Submodule.le_topologicalClosure _ (Submodule.subset_span ⟨t,hvj⟩) |
| have hi : inner ℂ v u=0 := (Submodule.mem_orthogonal _ u).mp hu v hvK |
| have he : inner ℂ v v=inner ℂ v u := by |
| rw [L2.inner_def,L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [hvE] with x hx |
| rw [hx] |
| by_cases hxE : x∈E <;> simp [Set.indicator_of_mem,Set.indicator_of_notMem,hxE] |
| have hv0 : v=0 := inner_self_eq_zero.mp (he.trans hi) |
| filter_upwards [hvE,Lp.coeFn_zero ℂ 2 μ] with x hx hx0 hxt |
| have hxE : x∈E := hxt |
| rw [Set.indicator_of_mem hxE,hv0,hx0] at hx |
| exact hx.symm |
| have hall : ∀ᵐ x ∂μ,∀ t∈S,r x=t → u x=0 := (ae_ball_iff hS).mpr (fun t _=>hz t) |
| have hu0 : u=0 := by |
| apply Lp.ext |
| filter_upwards [hall,hmem,Lp.coeFn_zero ℂ 2 μ] with x hx hxm hx0 |
| exact (hx (r x) hxm rfl).trans hx0.symm |
| simp only [hu0] |
| exact Submodule.orthogonal_eq_bot_iff.mp hK |
| end PolarStructure |
| end |
|
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| end OAI |
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