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f1f7953 | 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 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 | 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
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