| import OAI.Analysis.Brenier.MapDistance |
|
|
| namespace OAI |
|
|
| section |
|
|
| open MeasureTheory Filter |
| open scoped Topology BoundedContinuousFunction |
|
|
| namespace Problem358 |
|
|
| /-- Bounded continuous tests commute with almost-everywhere limits of measurable |
| maps on a finite measure space. No uniform bound on the maps themselves is needed. -/ |
| theorem tendsto_integral_boundedContinuous_comp_ae |
| {X Y : Type*} [MeasurableSpace X] [TopologicalSpace Y] |
| [MeasurableSpace Y] [OpensMeasurableSpace Y] |
| {ρ : Measure X} [IsFiniteMeasure ρ] |
| {Tn : ℕ → X → Y} {T : X → Y} |
| (hTn : ∀ n, Measurable (Tn n)) |
| (hlim : ∀ᵐ x ∂ρ, Tendsto (fun n => Tn n x) atTop (𝓝 (T x))) |
| (f : Y →ᵇ ℝ) : |
| Tendsto (fun n => ∫ x, f (Tn n x) ∂ρ) atTop |
| (𝓝 (∫ x, f (T x) ∂ρ)) := by |
| apply tendsto_integral_of_dominated_convergence (fun _ => ‖f‖) |
| · intro n |
| exact (f.continuous.measurable.comp (hTn n)).aestronglyMeasurable |
| · exact integrable_const _ |
| · intro n |
| exact Filter.Eventually.of_forall fun x => f.norm_coe_le_norm (Tn n x) |
| · filter_upwards [hlim] with x hx |
| exact f.continuous.continuousAt.tendsto.comp hx |
|
|
| /-- A candidate almost-everywhere limit has the prescribed target law whenever |
| the approximating laws converge against bounded continuous functions. -/ |
| theorem map_eq_of_ae_tendsto_and_test_tendsto |
| {X Y : Type*} [MeasurableSpace X] [TopologicalSpace Y] |
| [MeasurableSpace Y] [BorelSpace Y] [HasOuterApproxClosed Y] |
| {ρ : Measure X} [IsFiniteMeasure ρ] |
| {ν : Measure Y} [IsFiniteMeasure ν] |
| {Tn : ℕ → X → Y} {T : X → Y} |
| (hTn : ∀ n, Measurable (Tn n)) (hT : Measurable T) |
| (hlim : ∀ᵐ x ∂ρ, Tendsto (fun n => Tn n x) atTop (𝓝 (T x))) |
| (htest : ∀ f : Y →ᵇ ℝ, |
| Tendsto (fun n => ∫ x, f (Tn n x) ∂ρ) atTop (𝓝 (∫ y, f y ∂ν))) : |
| Measure.map T ρ = ν := by |
| apply ext_of_forall_integral_eq_of_IsFiniteMeasure |
| intro f |
| rw [integral_map hT.aemeasurable f.continuous.measurable.aestronglyMeasurable] |
| exact tendsto_nhds_unique (tendsto_integral_boundedContinuous_comp_ae hTn hlim f) (htest f) |
|
|
| /-- Closed support constraints survive almost-everywhere limits. -/ |
| theorem ae_mem_of_closed_of_ae_tendsto |
| {X Y : Type*} [MeasurableSpace X] [TopologicalSpace Y] |
| {ρ : Measure X} {Tn : ℕ → X → Y} {T : X → Y} {K : Set Y} |
| (hK : IsClosed K) (hmem : ∀ n, ∀ᵐ x ∂ρ, Tn n x ∈ K) |
| (hlim : ∀ᵐ x ∂ρ, Tendsto (fun n => Tn n x) atTop (𝓝 (T x))) : |
| ∀ᵐ x ∂ρ, T x ∈ K := by |
| have hall : ∀ᵐ x ∂ρ, ∀ n, Tn n x ∈ K := ae_all_iff.mpr hmem |
| filter_upwards [hall, hlim] with x hx htx |
| exact hK.mem_of_tendsto htx (Filter.Eventually.of_forall hx) |
|
|
| /-- The limit of maps to quantized target laws has the original target law. |
| This is the law-identification step in finite-target approximation. -/ |
| theorem map_eq_of_ae_tendsto_quantized |
| {X Y : Type*} [MeasurableSpace X] [TopologicalSpace Y] |
| [MeasurableSpace Y] [BorelSpace Y] [HasOuterApproxClosed Y] |
| {ρ : Measure X} [IsFiniteMeasure ρ] |
| {ν : Measure Y} [IsFiniteMeasure ν] |
| {Tn : ℕ → X → Y} {T : X → Y} {Qn : ℕ → Y → Y} |
| (hTn : ∀ n, Measurable (Tn n)) (hT : Measurable T) |
| (hQn : ∀ n, Measurable (Qn n)) |
| (hpush : ∀ n, Measure.map (Tn n) ρ = Measure.map (Qn n) ν) |
| (hlim : ∀ᵐ x ∂ρ, Tendsto (fun n => Tn n x) atTop (𝓝 (T x))) |
| (hQlim : ∀ᵐ y ∂ν, Tendsto (fun n => Qn n y) atTop (𝓝 y)) : |
| Measure.map T ρ = ν := by |
| apply map_eq_of_ae_tendsto_and_test_tendsto hTn hT hlim |
| intro f |
| have heq : ∀ n, (∫ x, f (Tn n x) ∂ρ) = ∫ y, f (Qn n y) ∂ν := by |
| intro n |
| rw [← integral_map (hTn n).aemeasurable f.continuous.measurable.aestronglyMeasurable, |
| hpush n, integral_map (hQn n).aemeasurable f.continuous.measurable.aestronglyMeasurable] |
| simp_rw [heq] |
| exact tendsto_integral_boundedContinuous_comp_ae hQn hQlim f |
|
|
| /-- Almost-everywhere convergence gives convergence against bounded continuous tests. -/ |
| theorem tendsto_integral_boundedContinuous_comp_of_ae |
| {X V : Type*} [MeasurableSpace X] [TopologicalSpace V] |
| [MeasurableSpace V] [BorelSpace V] |
| {ρ : Measure X} [IsFiniteMeasure ρ] |
| {Tn : ℕ → X → V} {T : X → V} (hTn : ∀ n, Measurable (Tn n)) |
| (hlim : ∀ᵐ x ∂ρ, Tendsto (fun n => Tn n x) atTop (𝓝 (T x))) |
| (f : BoundedContinuousFunction V ℝ) : |
| Tendsto (fun n => ∫ x, f (Tn n x) ∂ρ) atTop (𝓝 (∫ x, f (T x) ∂ρ)) := |
| tendsto_integral_boundedContinuous_comp_ae hTn hlim f |
|
|
| /-- Almost-everywhere limits with convergent approximating laws have the prescribed pushforward. -/ |
| theorem map_eq_of_ae_limits |
| {X V : Type*} [MeasurableSpace X] [TopologicalSpace V] |
| [MeasurableSpace V] [BorelSpace V] [HasOuterApproxClosed V] |
| {ρ : Measure X} {μ : Measure V} [IsFiniteMeasure ρ] [IsFiniteMeasure μ] |
| {Tn : ℕ → X → V} {T : X → V} {Qn : ℕ → V → V} |
| (hTn : ∀ n, Measurable (Tn n)) (hT : Measurable T) |
| (hQn : ∀ n, Measurable (Qn n)) |
| (hTlim : ∀ᵐ x ∂ρ, Tendsto (fun n => Tn n x) atTop (𝓝 (T x))) |
| (hQlim : ∀ᵐ y ∂μ, Tendsto (fun n => Qn n y) atTop (𝓝 y)) |
| (hpush : ∀ n, Measure.map (Tn n) ρ = Measure.map (Qn n) μ) : |
| Measure.map T ρ = μ := |
| map_eq_of_ae_tendsto_quantized hTn hT hQn hpush hTlim hQlim |
|
|
| end Problem358 |
|
|
| end |
|
|
| noncomputable section |
| open Filter MeasureTheory |
| open scoped Topology |
|
|
| namespace Problem358 |
|
|
| /-- Almost-everywhere existence of a pointwise limit with an arbitrary extra |
| property can be realized by one measurable function. No measurability of the |
| extra property is required. -/ |
| theorem exists_measurable_limit_with_property |
| {α β : Type*} [MeasurableSpace α] [MetricSpace β] |
| [MeasurableSpace β] [BorelSpace β] [Nonempty β] |
| {ρ : Measure α} {f : ℕ → α → β} {P : α → β → Prop} |
| (hmeas : ∀ n, AEMeasurable (f n) ρ) |
| (hex : ∀ᵐ x ∂ρ, ∃ y, Tendsto (fun n => f n x) atTop (𝓝 y) ∧ P x y) : |
| ∃ g : α → β, Measurable g ∧ ∀ᵐ x ∂ρ, |
| Tendsto (fun n => f n x) atTop (𝓝 (g x)) ∧ P x (g x) := by |
| classical |
| let Q : α → β → Prop := fun x y => |
| Tendsto (fun n => f n x) atTop (𝓝 y) ∧ P x y |
| let g : α → β := fun x => |
| if h : ∃ y, Q x y then Classical.choose h else Classical.arbitrary β |
| have hgprop : ∀ᵐ x ∂ρ, Q x (g x) := by |
| filter_upwards [hex] with x hx |
| dsimp only [g] |
| rw [dite_eq_left hx] |
| exact Classical.choose_spec hx |
| have hgmeas : AEMeasurable g ρ := |
| aemeasurable_of_tendsto_metrizable_ae atTop hmeas (hgprop.mono fun x hx => hx.1) |
| refine ⟨hgmeas.mk g, hgmeas.measurable_mk, ?_⟩ |
| filter_upwards [hgprop, hgmeas.ae_eq_mk] with x hx heq |
| exact heq ▸ hx |
|
|
| end Problem358 |
|
|
| end |
|
|
| noncomputable section |
|
|
| open MeasureTheory Filter |
| open scoped Topology ENNReal BoundedContinuousFunction |
|
|
| namespace Problem358 |
|
|
| variable {d : ℕ} {ρ ν : Measure (E d)} |
|
|
| /-- An L² limit has the target law when the approximating laws come from maps converging to |
| identity on that target. -/ |
| theorem map_eq_of_mapL2Dist_limit [IsFiniteMeasure ρ] [IsFiniteMeasure ν] |
| {T Q : ℕ → E d → E d} {S : E d → E d} |
| (hT : ∀ n, Measurable (T n)) (hQ : ∀ n, Measurable (Q n)) |
| (hS : Measurable S) (hTmem : ∀ n, MemLp (T n) 2 ρ) (hSmem : MemLp S 2 ρ) |
| (hTlim : Tendsto (fun n => mapL2Dist ρ (T n) S) atTop (𝓝 0)) |
| (hQlim : ∀ᵐ y ∂ν, Tendsto (fun n => Q n y) atTop (𝓝 y)) |
| (hmap : ∀ n, Measure.map (T n) ρ = Measure.map (Q n) ν) : |
| Measure.map S ρ = ν := by |
| obtain ⟨ns, hns, hlim⟩ := |
| exists_subseq_tendsto_ae_of_mapL2Dist_tendsto_zero hTmem hSmem hTlim |
| apply map_eq_of_ae_limits (fun n => hT (ns n)) hS (fun n => hQ (ns n)) hlim |
| _ (fun n => hmap (ns n)) |
| filter_upwards [hQlim] with y hy |
| exact hy.comp hns.tendsto_atTop |
|
|
| /-- Completeness and target-law identification combined, without an optimality assertion. -/ |
| theorem exists_map_of_approximating_mapL2_cauchy [IsFiniteMeasure ρ] [IsFiniteMeasure ν] |
| {T Q : ℕ → E d → E d} |
| (hT : ∀ n, Measurable (T n)) (hQ : ∀ n, Measurable (Q n)) |
| (hTmem : ∀ n, MemLp (T n) 2 ρ) |
| (hC : ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, |
| mapL2Dist ρ (T m) (T n) < ε) |
| (hQlim : ∀ᵐ y ∂ν, Tendsto (fun n => Q n y) atTop (𝓝 y)) |
| (hmap : ∀ n, Measure.map (T n) ρ = Measure.map (Q n) ν) : |
| ∃ S : E d → E d, Measurable S ∧ MemLp S 2 ρ ∧ Measure.map S ρ = ν ∧ |
| Tendsto (fun n => mapL2Dist ρ (T n) S) atTop (𝓝 0) := by |
| obtain ⟨S, hS, hSmem, hlim⟩ := exists_mapL2Dist_limit hTmem hC |
| exact ⟨S, hS, hSmem, map_eq_of_mapL2Dist_limit hT hQ hS hTmem hSmem hlim hQlim hmap, |
| hlim⟩ |
|
|
| end Problem358 |
|
|
| end |
|
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| end OAI |
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|
