openAI-math-dataset / lean /OAI /Probability /FreeIsing /Estimates /ContinuousCoefficientModulus.lean
| 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 | |