File size: 2,971 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 68 69 70 71 72 73 74 75 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory
open scoped symmDiff ENNReal
namespace Problem367
theorem exists_finite_coordinate_event
{ι : Type*} {X : ι → Type*} [∀ i, MeasurableSpace (X i)]
(μ : Measure (∀ i, X i)) [IsFiniteMeasure μ]
{A : Set (∀ i, X i)} (hA : MeasurableSet A)
{ε : ℝ≥0∞} (hε : 0 < ε) :
∃ (s : Finset ι) (B : Set (∀ i : s, X i)),
MeasurableSet B ∧ μ (cylinder s B ∆ A) < ε := by
obtain ⟨C, hC, hCA⟩ :=
exists_measure_symmDiff_lt_of_generateFrom_isSetRing
(μ := μ) isSetRing_measurableCylinders
(show ∃ D : Set (Set (∀ i, X i)), D.Countable ∧
D ⊆ measurableCylinders X ∧ μ (⋃₀ D)ᶜ = 0 from
⟨{Set.univ}, Set.countable_singleton _,
by simpa using univ_mem_measurableCylinders X, by simp⟩)
generateFrom_measurableCylinders.symm hA hε
obtain ⟨s, B, hB, rfl⟩ := (mem_measurableCylinders C).mp hC
exact ⟨s, B, hB, hCA⟩
theorem exists_finite_coordinate_bool
{ι : Type*} {X : ι → Type*} [∀ i, MeasurableSpace (X i)]
(μ : Measure (∀ i, X i)) [IsFiniteMeasure μ]
{f : (∀ i, X i) → Bool} (hf : Measurable f)
{ε : ℝ≥0∞} (hε : 0 < ε) :
∃ (s : Finset ι) (g : (∀ i : s, X i) → Bool),
Measurable g ∧ μ {x | g (s.restrict x) ≠ f x} < ε := by
classical
obtain ⟨s, B, hB, hBA⟩ := exists_finite_coordinate_event μ
(hf (measurableSet_singleton true)) hε
let g : (∀ i : s, X i) → Bool := fun x => if x ∈ B then true else false
refine ⟨s, g, Measurable.ite hB measurable_const measurable_const, ?_⟩
have hset : {x | g (s.restrict x) ≠ f x} =
cylinder s B ∆ (f ⁻¹' {true}) := by
ext x
cases hfx : f x <;> by_cases hx : s.restrict x ∈ B <;>
simp [g, Set.mem_symmDiff, cylinder, hfx, hx]
rwa [hset]
theorem exists_finite_coordinate_indicator_L2
{ι : Type*} {X : ι → Type*} [∀ i, MeasurableSpace (X i)]
(μ : Measure (∀ i, X i)) [IsFiniteMeasure μ]
{A : Set (∀ i, X i)} (hA : MeasurableSet A)
{ε : ℝ≥0∞} (hε : 0 < ε) :
∃ (s : Finset ι) (B : Set (∀ i : s, X i)),
MeasurableSet B ∧
eLpNorm ((cylinder s B).indicator (fun _ => (1 : ℝ)) -
A.indicator (fun _ => (1 : ℝ))) 2 μ < ε := by
obtain ⟨s, B, hB, hBA⟩ := exists_finite_coordinate_event μ hA (ENNReal.pow_pos hε 2)
refine ⟨s, B, hB, ?_⟩
rw [eLpNorm_indicator_sub_indicator _ hB.cylinder.nullMeasurableSet hA.nullMeasurableSet,
eLpNorm_indicator_const (hB.cylinder.symmDiff hA).nullMeasurableSet
(by norm_num) (by norm_num)]
simp only [enorm_one, one_mul, ENNReal.toReal_ofNat]
have h := ENNReal.rpow_lt_rpow hBA (by norm_num : (0 : ℝ) < 1 / 2)
have hpow : (ε ^ (2 : ℕ)) ^ (1 / 2 : ℝ) = ε := by
rw [← ENNReal.rpow_natCast, ← ENNReal.rpow_mul]
norm_num
rwa [hpow] at h
end Problem367
end
end OAI
|