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f61e547 | 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 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped Topology NNReal ENNReal
namespace Problem367.ContinuousCoefficientModulus
def pairModulus (x y : ℕ → ℝ) (n : ℕ) : ℝ :=
↑((Finset.range n).sup (fun i => ‖x i - y i‖₊))
theorem pairModulus_nonneg (x y : ℕ → ℝ) (n : ℕ) :
0 ≤ pairModulus x y n := NNReal.coe_nonneg _
theorem abs_sub_le_pairModulus (x y : ℕ → ℝ) {i n : ℕ} (hi : i < n) :
|x i - y i| ≤ pairModulus x y n := by
have h := Finset.le_sup (f := fun j => ‖x j - y j‖₊) (Finset.mem_range.mpr hi)
exact_mod_cast h
theorem pairModulus_le (x y : ℕ → ℝ) (n : ℕ) {C : ℝ} (hC : 0 ≤ C)
(hb : ∀ i < n, |x i - y i| ≤ C) : pairModulus x y n ≤ C := by
have h : (Finset.range n).sup (fun i => ‖x i - y i‖₊) ≤ (⟨C, hC⟩ : ℝ≥0) := by
apply Finset.sup_le
intro i hi
exact_mod_cast hb i (Finset.mem_range.mp hi)
exact_mod_cast h
variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
theorem stronglyMeasurable_pairModulus (x y : ℕ → Ω → ℝ)
(hx : ∀ i, StronglyMeasurable (x i)) (hy : ∀ i, StronglyMeasurable (y i)) (n : ℕ) :
StronglyMeasurable (fun ω => pairModulus (fun i => x i ω) (fun i => y i ω) n) := by
induction n with
| zero => simpa [pairModulus] using (stronglyMeasurable_const :
StronglyMeasurable (fun _ : Ω => (0 : ℝ)))
| succ n ih =>
have h := ((((hx n).measurable.sub (hy n).measurable).norm).max
ih.measurable).stronglyMeasurable
simpa only [pairModulus, Finset.range_add_one, Finset.sup_insert,
NNReal.coe_max, coe_nnnorm, Real.norm_eq_abs, Pi.sub_apply] using h
theorem memLp_pairModulus [IsFiniteMeasure μ] (x y : ℕ → Ω → ℝ)
(hx : ∀ i, StronglyMeasurable (x i)) (hy : ∀ i, StronglyMeasurable (y i))
(n : ℕ) {C : ℝ} (hC : 0 ≤ C)
(hb : ∀ i, i < n → ∀ᵐ ω ∂μ, |x i ω - y i ω| ≤ C) (p : ℝ≥0∞) :
MemLp (fun ω => pairModulus (fun i => x i ω) (fun i => y i ω) n) p μ := by
apply MemLp.of_bound (stronglyMeasurable_pairModulus x y hx hy n).aestronglyMeasurable C
have hb' : ∀ i, ∀ᵐ ω ∂μ, i < n → |x i ω - y i ω| ≤ C := by
intro i
by_cases hi : i < n
· exact (hb i hi).mono (fun _ h _ => h)
· exact Filter.Eventually.of_forall (fun _ h => (hi h).elim)
filter_upwards [ae_all_iff.2 hb'] with ω hω
rw [Real.norm_eq_abs, abs_of_nonneg (pairModulus_nonneg _ _ _)]
exact pairModulus_le _ _ _ hC hω
theorem tendsto_pairModulus_of_uniformContinuousOn
{I : Type*} [PseudoMetricSpace I] {S : Set I} {H : I → ℝ}
(hH : UniformContinuousOn H S) (s t : ℕ → ℕ → I) (N : ℕ → ℕ)
(hs : ∀ n i, i < N n → s n i ∈ S) (ht : ∀ n i, i < N n → t n i ∈ S)
(δ : ℕ → ℝ) (hδ : Tendsto δ atTop (𝓝 0))
(hdist : ∀ n i, i < N n → dist (s n i) (t n i) ≤ δ n) :
Tendsto (fun n => pairModulus (fun i => H (s n i)) (fun i => H (t n i)) (N n))
atTop (𝓝 0) := by
apply Metric.tendsto_atTop.mpr
intro ε hε
obtain ⟨η, hη, hlocal⟩ := Metric.uniformContinuousOn_iff.mp hH (ε / 2) (by positivity)
obtain ⟨n₀, hn₀⟩ := (hδ.eventually (gt_mem_nhds hη)).exists_forall_of_atTop
refine ⟨n₀, fun n hn => ?_⟩
have hb : pairModulus (fun i => H (s n i)) (fun i => H (t n i)) (N n) ≤ ε / 2 := by
apply pairModulus_le _ _ _ (by positivity)
intro i hi
have hh := hlocal (s n i) (hs n i hi) (t n i) (ht n i hi)
((hdist n i hi).trans_lt (hn₀ n hn))
simpa only [Real.dist_eq] using hh.le
rw [Real.dist_eq, sub_zero, abs_of_nonneg (pairModulus_nonneg _ _ _)]
linarith
theorem tendsto_integral_pairModulus_pow [IsFiniteMeasure μ]
{I : Type*} [PseudoMetricSpace I] {S : Set I} (hS : IsCompact S)
{H : I → Ω → ℝ} (hm : ∀ t, StronglyMeasurable (H t))
(hc : ∀ᵐ ω ∂μ, ContinuousOn (fun t => H t ω) S)
{C : ℝ} (hC : 0 ≤ C) (hb : ∀ t ∈ S, ∀ᵐ ω ∂μ, |H t ω| ≤ C)
(s t : ℕ → ℕ → I) (N : ℕ → ℕ)
(hs : ∀ n i, i < N n → s n i ∈ S) (ht : ∀ n i, i < N n → t n i ∈ S)
(δ : ℕ → ℝ) (hδ : Tendsto δ atTop (𝓝 0))
(hdist : ∀ n i, i < N n → dist (s n i) (t n i) ≤ δ n)
(p : ℕ) (hp : 0 < p) :
Tendsto (fun n => ∫ ω,
(pairModulus (fun i => H (s n i) ω) (fun i => H (t n i) ω) (N n)) ^ p ∂μ)
atTop (𝓝 0) := by
let R : ℕ → Ω → ℝ := fun n ω =>
pairModulus (fun i => H (s n i) ω) (fun i => H (t n i) ω) (N n)
have hRm (n : ℕ) : StronglyMeasurable (R n) :=
stronglyMeasurable_pairModulus _ _ (fun i => hm (s n i)) (fun i => hm (t n i)) (N n)
have hRb (n : ℕ) : ∀ᵐ ω ∂μ, |R n ω| ≤ 2 * C := by
have hb' : ∀ i, ∀ᵐ ω ∂μ, i < N n → |H (s n i) ω - H (t n i) ω| ≤ 2 * C := by
intro i
by_cases hi : i < N n
· filter_upwards [hb _ (hs n i hi), hb _ (ht n i hi)] with ω hsω htω
intro _
exact (abs_sub _ _).trans (by linarith)
· exact Filter.Eventually.of_forall (fun _ h => (hi h).elim)
filter_upwards [ae_all_iff.2 hb'] with ω hω
rw [abs_of_nonneg (pairModulus_nonneg _ _ _)]
exact pairModulus_le _ _ _ (by positivity) hω
have hlim : ∀ᵐ ω ∂μ, Tendsto (fun n => R n ω) atTop (𝓝 0) := by
filter_upwards [hc] with ω hω
exact tendsto_pairModulus_of_uniformContinuousOn
(hS.uniformContinuousOn_of_continuous hω) s t N hs ht δ hδ hdist
have hh := tendsto_integral_of_dominated_convergence (μ := μ)
(F := fun n ω => R n ω ^ p) (f := fun _ => (0 : ℝ))
(fun _ => (2 * C) ^ p) (fun n => (hRm n).aestronglyMeasurable.pow p)
(integrable_const _) ?_ ?_
· simpa only [integral_zero] using hh
· intro n
filter_upwards [hRb n] with ω hω
rw [norm_pow, Real.norm_eq_abs]
exact pow_le_pow_left₀ (abs_nonneg _) hω p
· filter_upwards [hlim] with ω hω
simpa only [zero_pow (Nat.ne_of_gt hp)] using hω.pow p
end Problem367.ContinuousCoefficientModulus
end
end OAI
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