openAI-math-dataset / lean /OAI /Analysis /JiangSu /DimensionFunctions /FiniteSpectralIntegralContinuity.lean
| import Mathlib | |
| import OAI.Analysis.TracialSplitting.PreparedBlocks | |
| import OAI.Analysis.JiangSu.DimensionFunctions.FiniteSpectralIntegral | |
| namespace OAI | |
| noncomputable section | |
| namespace Pureness | |
| open TraceCone MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology | |
| variable {D : Type*} [NonUnitalCStarAlgebra D] [PartialOrder D] [StarOrderedRing D] | |
| theorem positive_cut_continuous (ε : ℝ≥0) : Continuous (Positive.cut (A := D) ε) := by | |
| apply Continuous.subtype_mk | |
| apply continuous_iff_continuousAt.mpr | |
| intro M | |
| have hnorm : ∀ᶠ N : Positive D in 𝓝 M, ‖N.1‖ < ‖M.1‖+1 := | |
| continuous_subtype_val.norm.continuousAt.tendsto.eventually | |
| (gt_mem_nhds (lt_add_one ‖M.1‖)) | |
| have hs : ∀ᶠ N : Positive D in 𝓝 M, quasispectrum ℝ N.1 ⊆ Icc (-(‖M.1‖+1)) (‖M.1‖+1) := by | |
| filter_upwards [hnorm] with N hN | |
| intro t ht | |
| have h := NonUnitalIsometricContinuousFunctionalCalculus.norm_quasispectrum_le N.1 ht | |
| (IsSelfAdjoint.of_nonneg N.2) | |
| rw [Real.norm_eq_abs] at h | |
| exact abs_le.mp (h.trans hN.le) | |
| exact continuous_subtype_val.continuousAt.cfcₙ isCompact_Icc (fun t : ℝ => max (t-ε) 0) | |
| hs (Eventually.of_forall (fun N => IsSelfAdjoint.of_nonneg N.2)) | |
| variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] | |
| theorem finiteSpectralIntegral_lowerSemicontinuous | |
| (d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d) | |
| (hlsc : ∀ n, LowerSemicontinuous (fun M : Positive (FinMatrix A n) => d (finitePositiveClass n M))) | |
| (n : ℕ) : LowerSemicontinuous (finiteSpectralIntegral d n) := by | |
| apply lowerSemicontinuous_iff_le_liminf.mpr | |
| intro M | |
| calc | |
| finiteSpectralIntegral d n M ≤ ∫⁻ t in Ioi (0 : ℝ), | |
| liminf (fun N : Positive (FinMatrix A n) => finiteDimensionDistribution d n N t) (𝓝 M) := by | |
| apply lintegral_mono | |
| intro t | |
| exact ((hlsc n).comp (positive_cut_continuous t.toNNReal)).le_liminf M | |
| _ ≤ liminf (finiteSpectralIntegral d n) (𝓝 M) := | |
| lintegral_liminf_le (fun N => finiteDimensionDistribution_measurable d hd n N) | |
| theorem regularizedFiniteDimension_integral_lowerSemicontinuous | |
| (d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d) (n : ℕ) : | |
| LowerSemicontinuous (finiteSpectralIntegral (regularizedFiniteDimension d hd) n) := by | |
| apply finiteSpectralIntegral_lowerSemicontinuous _ (regularizedFiniteDimension_monotone d hd) | |
| intro m | |
| simpa only [regularizedFiniteDimension_eval] using finiteRegularizedValue_lowerSemicontinuous d hd m | |
| end Pureness | |
| end | |
| end OAI | |