| import Mathlib.MeasureTheory.Measure.Real |
| import Mathlib.Tactic |
|
|
| namespace OAI |
|
|
| open Filter MeasureTheory |
| open scoped Topology |
|
|
| namespace Balanced.Assembly |
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|
| theorem measureReal_le_good_add_three_errors {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsFiniteMeasure μ] (A E₁ E₂ E₃ : Set Ω) : |
| μ.real A ≤ μ.real (A \ (E₁ ∪ E₂ ∪ E₃)) + |
| (μ.real (A ∩ E₁) + μ.real (A ∩ E₂) + μ.real (A ∩ E₃)) := by |
| have hsub : A ⊆ (A \ (E₁ ∪ E₂ ∪ E₃)) ∪ ((A ∩ E₁) ∪ (A ∩ E₂) ∪ (A ∩ E₃)) := by |
| intro x hx |
| by_cases hbad : x ∈ E₁ ∪ E₂ ∪ E₃ |
| · right |
| rcases hbad with (h₁ | h₂) | h₃ |
| · exact Or.inl (Or.inl ⟨hx, h₁⟩) |
| · exact Or.inl (Or.inr ⟨hx, h₂⟩) |
| · exact Or.inr ⟨hx, h₃⟩ |
| · exact Or.inl ⟨hx, hbad⟩ |
| have hmono := measureReal_mono (μ := μ) hsub |
| have hsplit := measureReal_union_le (μ := μ) |
| (A \ (E₁ ∪ E₂ ∪ E₃)) ((A ∩ E₁) ∪ (A ∩ E₂) ∪ (A ∩ E₃)) |
| have h23 := measureReal_union_le (μ := μ) ((A ∩ E₁) ∪ (A ∩ E₂)) (A ∩ E₃) |
| have h12 := measureReal_union_le (μ := μ) (A ∩ E₁) (A ∩ E₂) |
| linarith |
|
|
| theorem positive_measureReal_good_of_three_errors_lt {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsFiniteMeasure μ] (A E₁ E₂ E₃ : Set Ω) |
| (h : μ.real (A ∩ E₁) + μ.real (A ∩ E₂) + μ.real (A ∩ E₃) < μ.real A) : |
| 0 < μ.real (A \ (E₁ ∪ E₂ ∪ E₃)) := by |
| have := measureReal_le_good_add_three_errors μ A E₁ E₂ E₃ |
| linarith |
|
|
| theorem eventually_positive_measureReal_good |
| {ι : Type*} {l : Filter ι} {Ω : ι → Type*} |
| [∀ i, MeasurableSpace (Ω i)] (μ : ∀ i, Measure (Ω i)) |
| [∀ i, IsFiniteMeasure (μ i)] (A E₁ E₂ E₃ : ∀ i, Set (Ω i)) |
| {c : ℝ} (hc : 0 < c) (hA : ∀ᶠ i in l, c ≤ (μ i).real (A i)) |
| (h₁ : Tendsto (fun i => (μ i).real (A i ∩ E₁ i)) l (𝓝 0)) |
| (h₂ : Tendsto (fun i => (μ i).real (A i ∩ E₂ i)) l (𝓝 0)) |
| (h₃ : Tendsto (fun i => (μ i).real (A i ∩ E₃ i)) l (𝓝 0)) : |
| ∀ᶠ i in l, 0 < (μ i).real (A i \ (E₁ i ∪ E₂ i ∪ E₃ i)) := by |
| have hsum : Tendsto (fun i => (μ i).real (A i ∩ E₁ i) + |
| (μ i).real (A i ∩ E₂ i) + (μ i).real (A i ∩ E₃ i)) l (𝓝 0) := by |
| simpa only [add_zero] using (h₁.add h₂).add h₃ |
| filter_upwards [hA, hsum.eventually_lt_const hc] with i hAi hi |
| exact positive_measureReal_good_of_three_errors_lt (μ i) (A i) (E₁ i) (E₂ i) (E₃ i) |
| (hi.trans_le hAi) |
|
|
| theorem eventually_exists_good |
| {ι : Type*} {l : Filter ι} {Ω : ι → Type*} |
| [∀ i, MeasurableSpace (Ω i)] (μ : ∀ i, Measure (Ω i)) |
| [∀ i, IsFiniteMeasure (μ i)] (A E₁ E₂ E₃ : ∀ i, Set (Ω i)) |
| {c : ℝ} (hc : 0 < c) (hA : ∀ᶠ i in l, c ≤ (μ i).real (A i)) |
| (h₁ : Tendsto (fun i => (μ i).real (A i ∩ E₁ i)) l (𝓝 0)) |
| (h₂ : Tendsto (fun i => (μ i).real (A i ∩ E₂ i)) l (𝓝 0)) |
| (h₃ : Tendsto (fun i => (μ i).real (A i ∩ E₃ i)) l (𝓝 0)) : |
| ∀ᶠ i in l, ∃ ω ∈ A i, ω ∉ E₁ i ∧ ω ∉ E₂ i ∧ ω ∉ E₃ i := by |
| filter_upwards [eventually_positive_measureReal_good μ A E₁ E₂ E₃ hc hA h₁ h₂ h₃] |
| with i hi |
| have hne : (A i \ (E₁ i ∪ E₂ i ∪ E₃ i)).Nonempty := by |
| by_contra h |
| have hempty := Set.not_nonempty_iff_eq_empty.mp h |
| simp only [hempty, measureReal_empty, lt_self_iff_false] at hi |
| rcases hne with ⟨ω, hω, hgood⟩ |
| exact ⟨ω, hω, by simpa only [Set.mem_union, not_or, and_assoc] using hgood⟩ |
|
|
| end Balanced.Assembly |
|
|
| end OAI |
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|
