File size: 2,956 Bytes
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 | 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
|