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ddc84c5 | 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 | import Mathlib.MeasureTheory.Measure.Portmanteau
import Mathlib.Topology.Algebra.Order.LiminfLimsup
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter
open scoped Topology ENNReal NNReal
/-- Compact subsets carrying uniformly almost all approximating mass carry
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]
/-- The loss outside a product compact set is bounded by its two marginal
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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