| import Mathlib |
| import OAI.Analysis.TracialSplitting.PreparedBlocks |
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|
| namespace OAI |
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| section |
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| noncomputable section |
| namespace Pureness |
| open Set TopologicalSpace MeasureTheory |
| open scoped NNReal ENNReal |
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| variable {X : Type*} [TopologicalSpace X] |
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| def dimensionCompactCapacity (r : C(X, ℝ≥0) → ℝ≥0∞) (K : Compacts X) : ℝ≥0∞ := |
| ⨅ (f : C(X, ℝ≥0)) (_ : ∀ x ∈ K, 1 ≤ f x), r f |
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| lemma dimensionCompactCapacity_le (r : C(X, ℝ≥0) → ℝ≥0∞) (K : Compacts X) |
| (f : C(X, ℝ≥0)) (hf : ∀ x ∈ K, 1 ≤ f x) : |
| dimensionCompactCapacity r K ≤ r f := iInf_le_of_le f (iInf_le_of_le hf le_rfl) |
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| lemma dimensionCompactCapacity_mono (r : C(X, ℝ≥0) → ℝ≥0∞) : |
| Monotone (dimensionCompactCapacity r) := by |
| intro K L hKL |
| exact le_iInf₂ fun f hf => dimensionCompactCapacity_le r K f (fun x hx => hf x (hKL hx)) |
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| lemma dimensionCompactCapacity_ne_top (r : C(X, ℝ≥0) → ℝ≥0∞) |
| (hr : r 1 ≠ ⊤) (K : Compacts X) : dimensionCompactCapacity r K ≠ ⊤ := |
| ne_top_of_le_ne_top hr (dimensionCompactCapacity_le r K 1 (fun _ _ => le_rfl)) |
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| lemma dimensionCompactCapacity_empty (r : C(X, ℝ≥0) → ℝ≥0∞) (hr : r 0 = 0) : |
| dimensionCompactCapacity r ⊥ = 0 := by |
| apply le_antisymm _ bot_le |
| exact (dimensionCompactCapacity_le r ⊥ 0 (by intro x hx; exact False.elim hx)).trans hr.le |
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| lemma dimensionCompactCapacity_sup_le (r : C(X, ℝ≥0) → ℝ≥0∞) |
| (hsub : ∀ f g, r (f+g) ≤ r f+r g) (K L : Compacts X) : |
| dimensionCompactCapacity r (K ⊔ L) ≤ dimensionCompactCapacity r K + dimensionCompactCapacity r L := by |
| unfold dimensionCompactCapacity |
| rw [ENNReal.iInf_add] |
| apply le_iInf |
| intro f |
| rw [ENNReal.iInf_add] |
| apply le_iInf |
| intro hf |
| rw [ENNReal.add_iInf] |
| apply le_iInf |
| intro g |
| rw [ENNReal.add_iInf] |
| apply le_iInf |
| intro hg |
| apply le_trans (dimensionCompactCapacity_le r (K ⊔ L) (f+g) ?_) (hsub f g) |
| intro x hx |
| rcases hx with hx | hx |
| · exact (hf x hx).trans (le_add_of_nonneg_right bot_le) |
| · exact (hg x hx).trans (le_add_of_nonneg_left bot_le) |
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|
| end Pureness |
| end |
| end |
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| end OAI |
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