| import Mathlib.MeasureTheory.Measure.Portmanteau |
| import Mathlib.Topology.Algebra.Order.LiminfLimsup |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter |
| open scoped Topology ENNReal NNReal |
|
|
| / |
| all limiting mass as their uniform losses tend to zero. -/ |
| theorem finiteMeasure_ae_of_compact_approximation {X : Type*} |
| [MeasurableSpace X] [TopologicalSpace X] [T2Space X] |
| [OpensMeasurableSpace X] [HasOuterApproxClosed X] |
| (μ : ℕ → FiniteMeasure X) (ν : FiniteMeasure X) |
| (hlim : Tendsto μ atTop (𝓝 ν)) (Z : Set X) (hZ : MeasurableSet Z) |
| (hcompact : ∀ ε : ℝ≥0, 0 < ε → ∃ K, IsCompact K ∧ K ⊆ Z ∧ |
| ∀ n, (μ n : Measure X) Kᶜ ≤ ε) : |
| ∀ᵐ x ∂(ν : Measure X), x ∈ Z := by |
| have hmass : Tendsto (fun n => (μ n : Measure X) Set.univ) atTop |
| (𝓝 ((ν : Measure X) Set.univ)) := by |
| simpa only [Function.comp_def, FiniteMeasure.ennreal_mass] using |
| (ENNReal.continuous_coe.tendsto ν.mass).comp hlim.mass |
| have hfull : (ν : Measure X) Set.univ ≤ (ν : Measure X) Z := by |
| apply ENNReal.le_of_forall_pos_le_add |
| intro ε hε _ |
| obtain ⟨K,hK,hKZ,hbound⟩ := hcompact ε hε |
| have hle : ∀ n, (μ n : Measure X) Set.univ ≤ (μ n : Measure X) K + ε := by |
| intro n |
| rw [← measure_add_measure_compl hK.isClosed.measurableSet] |
| exact add_le_add le_rfl (hbound n) |
| have hsup := Filter.limsup_le_limsup (f := atTop) (Eventually.of_forall hle) |
| rw [hmass.limsup_eq, limsup_add_const atTop _ _ (by isBoundedDefault) (by isBoundedDefault)] at hsup |
| exact hsup.trans (add_le_add |
| ((FiniteMeasure.limsup_measure_closed_le_of_tendsto hlim hK.isClosed).trans |
| (measure_mono hKZ)) le_rfl) |
| rw [ae_iff] |
| change (ν : Measure X) Zᶜ = 0 |
| rw [measure_compl hZ (by finiteness), tsub_eq_zero_of_le hfull] |
|
|
| / |
| losses. The identity set itself may only be Borel. -/ |
| theorem measure_pair_compact_loss {X : Type*} [MeasurableSpace X] |
| (μ : Measure (X × X)) (Q : Set X) (hQ : MeasurableSet Q) |
| (Z : Set (X × X)) (hZ : ∀ᵐ z ∂μ, z ∈ Z) : |
| μ ((Q ×ˢ Q) ∩ Z)ᶜ ≤ (μ.map Prod.fst) Qᶜ + (μ.map Prod.snd) Qᶜ := by |
| have he : (((Q ×ˢ Q) ∩ Z)ᶜ : Set (X × X)) =ᵐ[μ] ((Q ×ˢ Q)ᶜ : Set (X × X)) := by |
| filter_upwards [hZ] with z hz |
| apply propext |
| change (¬ (z ∈ Q ×ˢ Q ∧ z ∈ Z)) ↔ ¬ z ∈ Q ×ˢ Q |
| simp [hz] |
| rw [measure_congr he] |
| have hs : (Q ×ˢ Q)ᶜ = (Prod.fst ⁻¹' Qᶜ) ∪ (Prod.snd ⁻¹' Qᶜ) := by |
| ext z |
| simp only [Set.mem_compl_iff, Set.mem_prod, Set.mem_union, Set.mem_preimage] |
| tauto |
| rw [hs, Measure.map_apply measurable_fst hQ.compl, |
| Measure.map_apply measurable_snd hQ.compl] |
| exact measure_union_le _ _ |
|
|
| end FiniteFactor |
|
|
| end OAI |
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