openAI-math-dataset / lean /OAI /Analysis /JiangSu /DimensionFunctions /ContinuousDimensionIntegral.lean
| import Mathlib | |
| import OAI.Analysis.TracialSplitting.PreparedBlocks | |
| import OAI.Analysis.JiangSu.Model | |
| import OAI.Analysis.JiangSu.DimensionFunctions.DimensionContentMeasure | |
| namespace OAI | |
| noncomputable section | |
| namespace Pureness | |
| open Set TopologicalSpace MeasureTheory | |
| open scoped NNReal ENNReal | |
| variable {X : Type*} [TopologicalSpace X] | |
| def continuousDimensionIntegral (r : C(X, ℝ≥0) → ℝ≥0∞) (f : C(X, ℝ≥0)) : ℝ≥0∞ := | |
| ∫⁻ t in Ioi (0 : ℝ), r (continuousNNRealCut t.toNNReal f) | |
| variable [CompactSpace X] [T2Space X] [MeasurableSpace X] [BorelSpace X] | |
| theorem continuousDimensionIntegral_eq_lintegral | |
| (r : C(X, ℝ≥0) → ℝ≥0∞) (hfinite : r 1 ≠ ⊤) | |
| (hmono : Monotone r) (hsub : ∀ f g, r (f+g) ≤ r f+r g) | |
| (horth : ∀ f g, f*g = 0 → r (f+g) = r f+r g) | |
| (hsupport : ∀ f g, (∀ x, 0 < f x → 0 < g x) → r f ≤ r g) | |
| (hcuts : ∀ f, r f = ⨆ (ε : ℝ≥0) (_ : 0 < ε), r (continuousNNRealCut ε f)) | |
| (f : C(X, ℝ≥0)) : | |
| continuousDimensionIntegral r f = | |
| ∫⁻ x, (f x : ℝ≥0∞) ∂(dimensionContent r hfinite hmono hsub horth).measure := by | |
| let μ := (dimensionContent r hfinite hmono hsub horth).measure | |
| have hm : Measurable (fun x => (f x : ℝ)) := (NNReal.continuous_coe.comp f.continuous).measurable | |
| have layerCake : (∫⁻ point, (f point : ℝ≥0∞) ∂μ) = | |
| ∫⁻ level in Ioi (0 : ℝ), μ {point | level < (f point : ℝ)} := by | |
| calc | |
| _ = ∫⁻ point, ENNReal.ofReal (f point : ℝ) ∂μ := by | |
| apply lintegral_congr | |
| intro point | |
| exact ENNReal.ofReal_coe_nnreal.symm | |
| _ = _ := lintegral_eq_lintegral_meas_lt μ | |
| (Filter.Eventually.of_forall (fun point => (f point).2)) hm.aemeasurable | |
| rw [layerCake] | |
| apply setLIntegral_congr_fun measurableSet_Ioi | |
| intro t ht | |
| change r (continuousNNRealCut t.toNNReal f) = μ {x | t < (f x : ℝ)} | |
| rw [← dimensionContent_measure_support r hfinite hmono hsub horth hsupport hcuts] | |
| congr 1 | |
| ext x | |
| change 0 < f x-t.toNNReal ↔ t < (f x : ℝ) | |
| rw [tsub_pos_iff_lt] | |
| exact Real.toNNReal_lt_iff_lt_coe (show 0 ≤ t from le_of_lt ht) | |
| theorem continuousDimensionIntegral_add | |
| (r : C(X, ℝ≥0) → ℝ≥0∞) (hfinite : r 1 ≠ ⊤) | |
| (hmono : Monotone r) (hsub : ∀ f g, r (f+g) ≤ r f+r g) | |
| (horth : ∀ f g, f*g = 0 → r (f+g) = r f+r g) | |
| (hsupport : ∀ f g, (∀ x, 0 < f x → 0 < g x) → r f ≤ r g) | |
| (hcuts : ∀ f, r f = ⨆ (ε : ℝ≥0) (_ : 0 < ε), r (continuousNNRealCut ε f)) | |
| (f g : C(X, ℝ≥0)) : | |
| continuousDimensionIntegral r (f+g) = continuousDimensionIntegral r f + continuousDimensionIntegral r g := by | |
| rw [continuousDimensionIntegral_eq_lintegral r hfinite hmono hsub horth hsupport hcuts, | |
| continuousDimensionIntegral_eq_lintegral r hfinite hmono hsub horth hsupport hcuts, | |
| continuousDimensionIntegral_eq_lintegral r hfinite hmono hsub horth hsupport hcuts] | |
| simp only [ContinuousMap.add_apply,ENNReal.coe_add] | |
| exact lintegral_add_left (ENNReal.continuous_coe.comp f.continuous).measurable _ | |
| end Pureness | |
| end | |
| end OAI | |