| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter TopologicalSpace | |
| namespace Problem367 | |
| theorem ae_eq_of_dense_evaluations | |
| {Ω T E I : Type*} [MeasurableSpace Ω] {μ : Measure Ω} | |
| [TopologicalSpace T] [TopologicalSpace E] [T2Space E] [Countable I] | |
| (e : I → T) (he : DenseRange e) {f g : Ω → T → E} | |
| (hf : ∀ᵐ ω ∂μ, Continuous (f ω)) (hg : ∀ᵐ ω ∂μ, Continuous (g ω)) | |
| (hfg : ∀ i, ∀ᵐ ω ∂μ, f ω (e i) = g ω (e i)) : f =ᵐ[μ] g := by | |
| have hall : ∀ᵐ ω ∂μ, ∀ i, f ω (e i) = g ω (e i) := | |
| ae_all_iff.mpr hfg | |
| filter_upwards [hf, hg, hall] with ω hfω hgω hω | |
| exact he.equalizer hfω hgω (funext hω) | |
| theorem ae_eq_of_continuous_of_evaluation_ae_eq | |
| {Ω T E : Type*} [MeasurableSpace Ω] {μ : Measure Ω} | |
| [TopologicalSpace T] [SeparableSpace T] [Nonempty T] | |
| [TopologicalSpace E] [T2Space E] {f g : Ω → T → E} | |
| (hf : ∀ᵐ ω ∂μ, Continuous (f ω)) (hg : ∀ᵐ ω ∂μ, Continuous (g ω)) | |
| (hfg : ∀ t, ∀ᵐ ω ∂μ, f ω t = g ω t) : f =ᵐ[μ] g := by | |
| exact ae_eq_of_dense_evaluations (TopologicalSpace.denseSeq T) | |
| (TopologicalSpace.denseRange_denseSeq T) hf hg (fun _ => hfg _) | |
| theorem ae_eq_of_integral_sq_sub_eq_zero | |
| {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} {f g : Ω → ℝ} | |
| (hInt : Integrable (fun ω => (f ω - g ω) ^ 2) μ) | |
| (hzero : (∫ ω, (f ω - g ω) ^ 2 ∂μ) = 0) : f =ᵐ[μ] g := by | |
| have h := (integral_eq_zero_iff_of_nonneg (fun ω => sq_nonneg (f ω - g ω)) | |
| hInt).mp hzero | |
| filter_upwards [h] with ω hω | |
| exact sub_eq_zero.mp (sq_eq_zero_iff.mp hω) | |
| theorem ae_eq_of_continuous_of_integral_sq_sub_eq_zero | |
| {Ω T : Type*} [MeasurableSpace Ω] {μ : Measure Ω} | |
| [TopologicalSpace T] [SeparableSpace T] [Nonempty T] {f g : Ω → T → ℝ} | |
| (hf : ∀ᵐ ω ∂μ, Continuous (f ω)) (hg : ∀ᵐ ω ∂μ, Continuous (g ω)) | |
| (hInt : ∀ t, Integrable (fun ω => (f ω t - g ω t) ^ 2) μ) | |
| (hzero : ∀ t, (∫ ω, (f ω t - g ω t) ^ 2 ∂μ) = 0) : f =ᵐ[μ] g := by | |
| apply ae_eq_of_continuous_of_evaluation_ae_eq hf hg | |
| exact fun t => ae_eq_of_integral_sq_sub_eq_zero (hInt t) (hzero t) | |
| theorem ae_eq_array_of_continuous_of_integral_sq_sub_eq_zero | |
| {Ω V T : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [Countable V] | |
| [TopologicalSpace T] [SeparableSpace T] [Nonempty T] | |
| {f g : Ω → V → T → ℝ} | |
| (hf : ∀ v, ∀ᵐ ω ∂μ, Continuous (f ω v)) | |
| (hg : ∀ v, ∀ᵐ ω ∂μ, Continuous (g ω v)) | |
| (hInt : ∀ v t, Integrable (fun ω => (f ω v t - g ω v t) ^ 2) μ) | |
| (hzero : ∀ v t, (∫ ω, (f ω v t - g ω v t) ^ 2 ∂μ) = 0) : f =ᵐ[μ] g := by | |
| have h : ∀ v, ∀ᵐ ω ∂μ, f ω v = g ω v := fun v => | |
| ae_eq_of_continuous_of_integral_sq_sub_eq_zero (hf v) (hg v) (hInt v) (hzero v) | |
| filter_upwards [ae_all_iff.mpr h] with ω hω | |
| exact funext hω | |
| end Problem367 | |
| end | |
| end OAI | |