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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 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 | import Mathlib
import OAI.Analysis.TracialSplitting.PreparedBlocks
import OAI.Analysis.JiangSu.DimensionFunctions.DimensionContent
import OAI.Analysis.JiangSu.CStarAlgebras.ContinuousOrthogonalSeparation
namespace OAI
noncomputable section
namespace Pureness
open Set TopologicalSpace MeasureTheory
open scoped NNReal ENNReal
variable {X : Type*} [TopologicalSpace X] [T2Space X] [NormalSpace X]
lemma dimensionCompactCapacity_sup_disjoint (r : C(X, ℝ≥0) → ℝ≥0∞)
(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)
(K L : Compacts X) (hKL : Disjoint (K : Set X) L) :
dimensionCompactCapacity r (K ⊔ L) = dimensionCompactCapacity r K + dimensionCompactCapacity r L := by
apply le_antisymm (dimensionCompactCapacity_sup_le r hsub K L)
obtain ⟨u,v,hu,hv,hu1,hv1,huv⟩ := continuous_nnreal_orthogonal_separators K.isCompact.isClosed L.isCompact.isClosed hKL
apply le_iInf₂
intro f hf
have hfu : ∀ x ∈ K, 1 ≤ (f*u) x := by
intro x hx
simpa only [ContinuousMap.mul_apply,hu x hx,mul_one] using hf x (Or.inl hx)
have hfv : ∀ x ∈ L, 1 ≤ (f*v) x := by
intro x hx
simpa only [ContinuousMap.mul_apply,hv x hx,mul_one] using hf x (Or.inr hx)
have huv1 : u+v ≤ 1 := by
intro x
change u x+v x ≤ 1
have hz : u x*v x = 0 := congrArg (fun g : C(X, ℝ≥0) => g x) huv
rcases mul_eq_zero.mp hz with hz | hz
· rw [hz,zero_add]
exact hv1 x
· rw [hz,add_zero]
exact hu1 x
have horth' : (f*u)*(f*v) = 0 := by
calc
(f*u)*(f*v) = (f*f)*(u*v) := by ring
_ = 0 := by rw [huv,mul_zero]
calc
dimensionCompactCapacity r K + dimensionCompactCapacity r L ≤ r (f*u)+r (f*v) :=
add_le_add (dimensionCompactCapacity_le r K (f*u) hfu) (dimensionCompactCapacity_le r L (f*v) hfv)
_ = r (f*u+f*v) := (horth (f*u) (f*v) horth').symm
_ ≤ r f := hmono (by
intro x
change f x*u x+f x*v x ≤ f x
have h := mul_le_mul_of_nonneg_left (huv1 x) (show 0 ≤ f x from bot_le)
change f x*(u x+v x) ≤ f x*1 at h
simpa only [mul_add,mul_one] using h)
def dimensionContent (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) : Content X where
toFun K := (dimensionCompactCapacity r K).toNNReal
mono' K L h := ENNReal.toNNReal_mono (dimensionCompactCapacity_ne_top r hfinite L)
(dimensionCompactCapacity_mono r h)
sup_disjoint' K L h _ _ := by
rw [dimensionCompactCapacity_sup_disjoint r hmono hsub horth K L h,
ENNReal.toNNReal_add (dimensionCompactCapacity_ne_top r hfinite K) (dimensionCompactCapacity_ne_top r hfinite L)]
sup_le' K L := by
rw [← ENNReal.toNNReal_add (dimensionCompactCapacity_ne_top r hfinite K) (dimensionCompactCapacity_ne_top r hfinite L)]
exact ENNReal.toNNReal_mono (ENNReal.add_ne_top.mpr ⟨dimensionCompactCapacity_ne_top r hfinite K,
dimensionCompactCapacity_ne_top r hfinite L⟩) (dimensionCompactCapacity_sup_le r hsub K L)
lemma dimensionContent_apply (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) (K : Compacts X) :
dimensionContent r hfinite hmono hsub horth K = dimensionCompactCapacity r K :=
ENNReal.coe_toNNReal (dimensionCompactCapacity_ne_top r hfinite K)
end Pureness
namespace Pureness
open Set TopologicalSpace MeasureTheory
open scoped NNReal ENNReal
variable {X : Type*} [TopologicalSpace X]
def continuousNNRealCut (ε : ℝ≥0) (f : C(X, ℝ≥0)) : C(X, ℝ≥0) :=
⟨fun x => f x-ε, f.continuous.sub continuous_const⟩
variable [CompactSpace X] [T2Space X] [MeasurableSpace X] [BorelSpace X]
theorem dimensionContent_measure_support
(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)) :
(dimensionContent r hfinite hmono hsub horth).measure {x | 0 < f x} = r f := by
let c := dimensionContent r hfinite hmono hsub horth
have hU : IsOpen {x | 0 < f x} := isOpen_lt continuous_const f.continuous
rw [Content.measure_apply _ hU.measurableSet,Content.outerMeasure_of_isOpen _ _ hU]
apply le_antisymm
· apply iSup₂_le
intro K hK
obtain ⟨g,hgK,hgc,hgU,hg⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen K.isCompact hU hK
let g' : C(X, ℝ≥0) := ⟨fun x => (g x).toNNReal, continuous_real_toNNReal.comp g.continuous⟩
have hg'K : ∀ x ∈ K, 1 ≤ g' x := by
intro x hx
change 1 ≤ (g x).toNNReal
rw [hgK hx]
norm_num
rw [dimensionContent_apply]
apply (dimensionCompactCapacity_le r K g' hg'K).trans
apply hsupport
intro x hx
apply hgU
apply subset_closure
change g x ≠ 0
have hp : 0 < g x := Real.toNNReal_pos.mp hx
exact hp.ne'
· rw [hcuts f]
apply iSup₂_le
intro ε hε
let K : Compacts X := ⟨{x | ε ≤ f x}, (isClosed_le continuous_const f.continuous).isCompact⟩
have hKU : (K : Set X) ⊆ {x | 0 < f x} := fun x hx => hε.trans_le hx
apply le_trans _ (c.le_innerContent K ⟨_,hU⟩ hKU)
rw [dimensionContent_apply]
apply le_iInf₂
intro g hg
apply hsupport
intro x hx
have hxε : ε < f x := tsub_pos_iff_lt.mp hx
exact zero_lt_one.trans_le (hg x hxε.le)
end Pureness
end
end OAI
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