| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory Filter |
| open scoped Topology |
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| namespace Problem367 |
| namespace ContinuousModification |
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| variable {T Ω : Type*} |
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| def sampledLimit (q : T → ℕ → T) (F : T → Ω → ℝ) (t : T) (ω : Ω) : ℝ := |
| limUnder atTop (fun n => F (q t n) ω) |
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| theorem sampledLimit_congr {q : T → ℕ → T} {F G : T → Ω → ℝ} |
| {t : T} {ω ω' : Ω} (h : ∀ n, F (q t n) ω = G (q t n) ω') : |
| sampledLimit q F t ω = sampledLimit q G t ω' := by |
| unfold sampledLimit |
| congr 1 |
| exact funext h |
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| theorem measurable_sampledLimit [MeasurableSpace T] [MeasurableSpace Ω] |
| {q : T → ℕ → T} {F : T → Ω → ℝ} |
| (hF : ∀ n, Measurable (fun z : T × Ω => F (q z.1 n) z.2)) : |
| Measurable (fun z : T × Ω => sampledLimit q F z.1 z.2) := by |
| exact (StronglyMeasurable.limUnder (fun n => (hF n).stronglyMeasurable)).measurable |
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| theorem measurable_sampledLimit_at [MeasurableSpace Ω] |
| {q : T → ℕ → T} {F : T → Ω → ℝ} (t : T) |
| (hF : ∀ n, Measurable (F (q t n))) : Measurable (sampledLimit q F t) := by |
| exact (StronglyMeasurable.limUnder (fun n => (hF n).stronglyMeasurable)).measurable |
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| theorem ae_all_eq_of_continuous_modification [TopologicalSpace T] [MeasurableSpace Ω] |
| {μ : Measure Ω} {q : T → ℕ → T} {F G : T → Ω → ℝ} |
| {S : Set T} (hS : S.Countable) |
| (hqS : ∀ t n, q t n ∈ S) |
| (hq : ∀ t, Tendsto (q t) atTop (𝓝 t)) |
| (hFG : ∀ s ∈ S, F s =ᵐ[μ] G s) |
| (hG : ∀ᵐ ω ∂μ, Continuous (fun t => G t ω)) : |
| ∀ᵐ ω ∂μ, ∀ t, sampledLimit q F t ω = G t ω := by |
| let : Countable S := hS.to_subtype |
| have hEq : ∀ᵐ ω ∂μ, ∀ s : S, F s ω = G s ω := |
| ae_all_iff.mpr (fun s => hFG s s.property) |
| filter_upwards [hEq, hG] with ω hEqω hGω |
| intro t |
| have hseq : (fun n => F (q t n) ω) = (fun n => G (q t n) ω) := |
| funext (fun n => hEqω ⟨q t n, hqS t n⟩) |
| unfold sampledLimit |
| rw [hseq] |
| exact ((hGω.tendsto t).comp (hq t)).limUnder_eq |
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| theorem ae_continuous_sampledLimit [TopologicalSpace T] [MeasurableSpace Ω] |
| {μ : Measure Ω} {q : T → ℕ → T} {F G : T → Ω → ℝ} |
| {S : Set T} (hS : S.Countable) |
| (hqS : ∀ t n, q t n ∈ S) |
| (hq : ∀ t, Tendsto (q t) atTop (𝓝 t)) |
| (hFG : ∀ s ∈ S, F s =ᵐ[μ] G s) |
| (hG : ∀ᵐ ω ∂μ, Continuous (fun t => G t ω)) : |
| ∀ᵐ ω ∂μ, Continuous (fun t => sampledLimit q F t ω) := by |
| filter_upwards [ae_all_eq_of_continuous_modification hS hqS hq hFG hG, hG] |
| with ω hEq hGω |
| simpa only [hEq] using hGω |
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| theorem sampledLimit_equivariant {I : Type*} {q : T → ℕ → T} |
| (F : I → T → Ω → ℝ) (a : Ω → Ω) (e : I → I) |
| (hF : ∀ i t n ω, F i (q t n) (a ω) = F (e i) (q t n) ω) : |
| ∀ i t ω, sampledLimit q (F i) t (a ω) = sampledLimit q (F (e i)) t ω := by |
| intro i t ω |
| exact sampledLimit_congr (fun n => hF i t n ω) |
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| theorem ae_all_eq_family [TopologicalSpace T] [MeasurableSpace Ω] |
| {I : Type*} [Countable I] {μ : Measure Ω} {q : T → ℕ → T} |
| {F G : I → T → Ω → ℝ} {S : Set T} (hS : S.Countable) |
| (hqS : ∀ t n, q t n ∈ S) |
| (hq : ∀ t, Tendsto (q t) atTop (𝓝 t)) |
| (hFG : ∀ i s, s ∈ S → F i s =ᵐ[μ] G i s) |
| (hG : ∀ i, ∀ᵐ ω ∂μ, Continuous (fun t => G i t ω)) : |
| ∀ᵐ ω ∂μ, ∀ i t, sampledLimit q (F i) t ω = G i t ω := by |
| apply ae_all_iff.mpr |
| intro i |
| exact ae_all_eq_of_continuous_modification hS hqS hq (hFG i) (hG i) |
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| theorem isStronglyProgressive_sampledLimit [Preorder T] [MeasurableSpace T] |
| [mΩ : MeasurableSpace Ω] (ℱ : Filtration T mΩ) |
| {q : T → ℕ → T} {F : T → Ω → ℝ} |
| (hF : ∀ n, IsStronglyProgressive ℱ (fun t ω => F (q t n) ω)) : |
| IsStronglyProgressive ℱ (sampledLimit q F) := by |
| intro t |
| let : MeasurableSpace Ω := ℱ t |
| exact StronglyMeasurable.limUnder (fun n => hF n t) |
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| def dyadicLeft (t : ℝ) (n : ℕ) : ℝ := (⌊(2 : ℝ)^n * t⌋ : ℤ) / (2 : ℝ)^n |
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| theorem dyadicLeft_le (t : ℝ) (n : ℕ) : dyadicLeft t n ≤ t := by |
| unfold dyadicLeft |
| apply (div_le_iff₀ (by positivity : (0 : ℝ) < 2^n)).2 |
| simpa only [mul_comm] using Int.floor_le ((2 : ℝ)^n * t) |
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| theorem sub_inv_pow_le_dyadicLeft (t : ℝ) (n : ℕ) : |
| t - ((2 : ℝ)^n)⁻¹ ≤ dyadicLeft t n := by |
| have hp : (0 : ℝ) < 2^n := by positivity |
| unfold dyadicLeft |
| apply (le_div_iff₀ hp).2 |
| have hfloor := Int.lt_floor_add_one ((2 : ℝ)^n * t) |
| have hinv : ((2 : ℝ)^n)⁻¹ * (2 : ℝ)^n = 1 := inv_mul_cancel₀ hp.ne' |
| nlinarith |
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| theorem dyadicLeft_nonneg {t : ℝ} (ht : 0 ≤ t) (n : ℕ) : |
| 0 ≤ dyadicLeft t n := by |
| unfold dyadicLeft |
| exact div_nonneg (Int.cast_nonneg (Int.floor_nonneg.mpr (by positivity))) (by positivity) |
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| theorem tendsto_dyadicLeft (t : ℝ) : Tendsto (dyadicLeft t) atTop (𝓝 t) := by |
| have hinv : Tendsto (fun n : ℕ => ((2 : ℝ)^n)⁻¹) atTop (𝓝 0) := |
| tendsto_inv_atTop_zero.comp (tendsto_pow_atTop_atTop_of_one_lt (by norm_num)) |
| have hlow : Tendsto (fun n : ℕ => t - ((2 : ℝ)^n)⁻¹) atTop (𝓝 t) := by |
| simpa using tendsto_const_nhds.sub hinv |
| exact tendsto_of_tendsto_of_tendsto_of_le_of_le hlow tendsto_const_nhds |
| (sub_inv_pow_le_dyadicLeft t) (dyadicLeft_le t) |
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| theorem dyadicLeft_mem_rationals (t : ℝ) (n : ℕ) : |
| dyadicLeft t n ∈ Set.range (fun r : ℚ => (r : ℝ)) := by |
| refine ⟨(⌊(2 : ℝ)^n * t⌋ : ℚ) / (2 : ℚ)^n, ?_⟩ |
| simp only [Rat.cast_div, Rat.cast_intCast, Rat.cast_pow, Rat.cast_ofNat, dyadicLeft] |
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| theorem measurable_dyadic_sample [MeasurableSpace Ω] {F : ℝ → Ω → ℝ} |
| (hF : ∀ t, Measurable (F t)) (n : ℕ) : |
| Measurable (fun z : ℝ × Ω => F (dyadicLeft z.1 n) z.2) := by |
| have hstep : Measurable (fun z : ℤ × Ω => F ((z.1 : ℝ) / (2 : ℝ)^n) z.2) := |
| measurable_from_prod_countable_right (fun k => hF ((k : ℝ) / (2 : ℝ)^n)) |
| exact hstep.comp (((measurable_const.mul measurable_fst).floor).prodMk measurable_snd) |
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| theorem isProgressive_dyadic_sample [mΩ : MeasurableSpace Ω] |
| (ℱ : Filtration ℝ mΩ) {F : ℝ → Ω → ℝ} |
| (hF : Adapted ℱ F) (n : ℕ) : |
| IsProgressive ℱ (fun t ω => F (dyadicLeft t n) ω) := by |
| intro t |
| let : MeasurableSpace Ω := ℱ t |
| have hstep : Measurable |
| (fun z : ℤ × Ω => F (min ((z.1 : ℝ) / (2 : ℝ)^n) t) z.2) := |
| measurable_from_prod_countable_right (fun k => by |
| exact (hF (min ((k : ℝ) / (2 : ℝ)^n) t)).mono |
| (ℱ.mono (min_le_right _ _)) le_rfl) |
| have hfloor : Measurable |
| (fun z : Set.Iic t × Ω => (⌊(2 : ℝ)^n * (z.1 : ℝ)⌋, z.2)) := |
| ((measurable_const.mul (measurable_subtype_coe.comp measurable_fst)).floor).prodMk |
| measurable_snd |
| have heq : (fun z : Set.Iic t × Ω => F (dyadicLeft (z.1 : ℝ) n) z.2) = |
| (fun z : Set.Iic t × Ω => F (min (dyadicLeft (z.1 : ℝ) n) t) z.2) := by |
| funext z |
| rw [min_eq_left ((dyadicLeft_le (z.1 : ℝ) n).trans z.1.property)] |
| rw [heq] |
| exact hstep.comp hfloor |
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| theorem isStronglyProgressive_dyadicLimit [mΩ : MeasurableSpace Ω] |
| (ℱ : Filtration ℝ mΩ) {F : ℝ → Ω → ℝ} (hF : Adapted ℱ F) : |
| IsStronglyProgressive ℱ (sampledLimit dyadicLeft F) := |
| isStronglyProgressive_sampledLimit ℱ |
| (fun n => (isProgressive_dyadic_sample ℱ hF n).isStronglyProgressive) |
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| theorem measurable_dyadicLimit [MeasurableSpace Ω] {F : ℝ → Ω → ℝ} |
| (hF : ∀ t, Measurable (F t)) : |
| Measurable (fun z : ℝ × Ω => sampledLimit dyadicLeft F z.1 z.2) := |
| measurable_sampledLimit (measurable_dyadic_sample hF) |
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| theorem ae_all_dyadicLimit_eq [MeasurableSpace Ω] {μ : Measure Ω} |
| {F G : ℝ → Ω → ℝ} |
| (hFG : ∀ r : ℚ, F (r : ℝ) =ᵐ[μ] G (r : ℝ)) |
| (hG : ∀ᵐ ω ∂μ, Continuous (fun t => G t ω)) : |
| ∀ᵐ ω ∂μ, ∀ t, sampledLimit dyadicLeft F t ω = G t ω := by |
| apply ae_all_eq_of_continuous_modification |
| (Set.countable_range (fun r : ℚ => (r : ℝ))) dyadicLeft_mem_rationals tendsto_dyadicLeft |
| ?_ hG |
| rintro s ⟨r, rfl⟩ |
| exact hFG r |
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| end ContinuousModification |
| end Problem367 |
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| end |
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| end OAI |
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