openAI-math-dataset / lean /OAI /Analysis /JiangSu /DimensionFunctions /DimensionContentMeasure.lean
| 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 | |