AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge

- Extend a function on nonnegative time by zero to the negative real axis.

noncomputable def realZeroExtension (f : ℝ≥0 → ℝ) : ℝ → ℝ :=
  fun r ↦ if 0 ≤ r then f r.toNNReal else 0

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:27published source at 7bcd37294df1

No declaration docstring.

theorem realZeroExtension_measurable {f : ℝ≥0 → ℝ} (hf : Measurable f) :
    Measurable (realZeroExtension f) := by
  exact (hf.comp measurable_real_toNNReal).piecewise
    measurableSet_Ici measurable_const

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:30published source at 7bcd37294df1

No declaration docstring.

theorem realZeroExtension_stronglyMeasurable
    {f : ℝ≥0 → ℝ} (hf : StronglyMeasurable f) :
    StronglyMeasurable (realZeroExtension f) :=
  (realZeroExtension_measurable hf.measurable).stronglyMeasurable

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:35published source at 7bcd37294df1

No declaration docstring.

@[simp] theorem realZeroExtension_coe (f : ℝ≥0 → ℝ) (t : ℝ≥0) :
    realZeroExtension f (t : ℝ) = f t := by
  simp [realZeroExtension]

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:40published source at 7bcd37294df1

No declaration docstring.

theorem realZeroExtension_eq_zero_of_neg (f : ℝ≥0 → ℝ)
    {r : ℝ} (hr : r < 0) :
    realZeroExtension f r = 0 := by
  simp [realZeroExtension, not_le.mpr hr]

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:44published source at 7bcd37294df1

No declaration docstring.

private theorem range_nnreal_coe :
    Set.range ((↑) : ℝ≥0 → ℝ) = Ici 0 := by
  ext r
  constructor
  · rintro ⟨s, rfl⟩
    exact s.property
  · intro hr
    exact ⟨⟨r, hr⟩, rfl⟩

/-- On an interval contained in `[0,T]`, pushing `upTo T` forward along the
canonical embedding gives ordinary Lebesgue measure restricted to that real
interval. -/

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:49published source at 7bcd37294df1

- On an interval contained in `[0,T]`, pushing `upTo T` forward along the canonical embedding gives ordinary Lebesgue measure restricted to that real interval.

theorem map_restrict_upTo_Ioc
    {a b T : ℝ≥0} (_hab : a ≤ b) (hbT : b ≤ T) :
    Measure.map ((↑) : ℝ≥0 → ℝ)
        ((TimeMeasure.upTo T).restrict (Ioc a b)) =
      volume.restrict (Ioc (a : ℝ) (b : ℝ)) := by
  let emb : ℝ≥0 → ℝ := (↑)
  have hemb : MeasurableEmbedding emb :=
    MeasurableEmbedding.subtype_coe measurableSet_Ici
  rw [TimeMeasure.upTo, hemb.restrict_comap, hemb.map_comap]
  rw [NNReal.image_coe_Ioc, range_nnreal_coe]
  rw [Measure.restrict_restrict measurableSet_Ici]
  rw [Measure.restrict_restrict
    (measurableSet_Ici.inter (measurableSet_Ioc : MeasurableSet (Ioc (a : ℝ) b)))]
  congr 1
  ext r
  simp only [mem_inter_iff, mem_Icc, mem_Ioc, mem_Ici]
  constructor
  · rintro ⟨⟨hr0, har, hrb⟩, _, hrT⟩
    exact ⟨har, hrb⟩
  · rintro ⟨har, hrb⟩
    have hr0 : 0 ≤ r := le_trans a.property har.le
    have hrT : r ≤ (T : ℝ) := hrb.trans (by exact_mod_cast hbT)
    exact ⟨⟨hr0, har, hrb⟩, hr0, hrT⟩

/-- Exact Bochner-integral bridge on `(a,b]`.  No endpoint regularity is
assumed; the interval convention agrees on both sides. -/

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:61published source at 7bcd37294df1

- Exact Bochner-integral bridge on `(a,b]`. No endpoint regularity is assumed; the interval convention agrees on both sides.

theorem integral_upTo_restrict_Ioc_eq_real
    (f : ℝ≥0 → ℝ) {a b T : ℝ≥0}
    (hab : a ≤ b) (hbT : b ≤ T) :
    ∫ s, f s ∂((TimeMeasure.upTo T).restrict (Ioc a b)) =
      ∫ r in (a : ℝ)..(b : ℝ), realZeroExtension f r := by
  have hemb : MeasurableEmbedding ((↑) : ℝ≥0 → ℝ) :=
    MeasurableEmbedding.subtype_coe measurableSet_Ici
  rw [intervalIntegral.integral_of_le (by exact_mod_cast hab)]
  rw [← map_restrict_upTo_Ioc hab hbT]
  calc
    ∫ s, f s ∂((TimeMeasure.upTo T).restrict (Ioc a b)) =
        ∫ s, realZeroExtension f (s : ℝ)
          ∂((TimeMeasure.upTo T).restrict (Ioc a b)) := by
      apply integral_congr_ae
      filter_upwards [] with s
      exact (realZeroExtension_coe f s).symm
    _ = ∫ r, realZeroExtension f r
          ∂Measure.map ((↑) : ℝ≥0 → ℝ)
            ((TimeMeasure.upTo T).restrict (Ioc a b)) :=
      (hemb.integral_map (realZeroExtension f)).symm

/-- An a.e. statement on a nonnegative interval is equivalent to its real
zero-coordinate form under ordinary restricted Lebesgue measure. -/

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:87published source at 7bcd37294df1

- An a.e. statement on a nonnegative interval is equivalent to its real zero-coordinate form under ordinary restricted Lebesgue measure.

theorem ae_restrict_upTo_Ioc_iff_real
    (p : ℝ≥0 → Prop) {a b T : ℝ≥0}
    (hab : a ≤ b) (hbT : b ≤ T) :
    (∀ᵐ s ∂((TimeMeasure.upTo T).restrict (Ioc a b)), p s) ↔
      ∀ᵐ r ∂(volume.restrict (Ioc (a : ℝ) (b : ℝ))), p r.toNNReal := by
  have hemb : MeasurableEmbedding ((↑) : ℝ≥0 → ℝ) :=
    MeasurableEmbedding.subtype_coe measurableSet_Ici
  have hmap := hemb.ae_map_iff
    (p := fun r : ℝ ↦ p r.toNNReal)
    (μ := (TimeMeasure.upTo T).restrict (Ioc a b))
  rw [map_restrict_upTo_Ioc hab hbT] at hmap
  simpa using hmap.symm

/-- Upgrade pointwise-in-`omega` time-a.e. facts to product-a.e. facts once
the target event is known measurable. -/

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:110published source at 7bcd37294df1

- Upgrade pointwise-in-`omega` time-a.e. facts to product-a.e. facts once the target event is known measurable.

theorem ae_prod_restrict_upTo_of_forall_ae
    {p : Omega × ℝ≥0 → Prop} {a b T : ℝ≥0}
    (hp : MeasurableSet {z | p z})
    (h : ∀ omega, ∀ᵐ s ∂((TimeMeasure.upTo T).restrict (Ioc a b)), p (omega, s)) :
    ∀ᵐ z ∂mu.prod ((TimeMeasure.upTo T).restrict (Ioc a b)), p z := by
  apply (Measure.ae_prod_iff_ae_ae hp).2
  filter_upwards [] with omega
  exact h omega

/-- Real-time section of the clipped process, zero outside `[0,T]`. -/

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:125published source at 7bcd37294df1

- Real-time section of the clipped process, zero outside `[0,T]`.

noncomputable def realClippedSection
    (eta : ProgressiveL2Integrand filtration mu T)
    (truncationLevel : ℕ) (omega : Omega) : ℝ → ℝ :=
  realZeroExtension
    (fun s ↦ clippedExtensionAt eta truncationLevel T (s, omega))

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:135published source at 7bcd37294df1

No declaration docstring.

@[simp] theorem realClippedSection_coe
    (eta : ProgressiveL2Integrand filtration mu T)
    (truncationLevel : ℕ) (omega : Omega) {s : ℝ≥0} (hsT : s ≤ T) :
    realClippedSection eta truncationLevel omega (s : ℝ) =
      (clipped eta truncationLevel).process s omega := by
  simp [realClippedSection,
    clippedExtensionAt_apply_of_le eta truncationLevel hsT]

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:141published source at 7bcd37294df1

No declaration docstring.

theorem realClippedSection_eq_zero_of_neg
    (eta : ProgressiveL2Integrand filtration mu T)
    (truncationLevel : ℕ) (omega : Omega) {r : ℝ} (hr : r < 0) :
    realClippedSection eta truncationLevel omega r = 0 :=
  realZeroExtension_eq_zero_of_neg _ hr

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:149published source at 7bcd37294df1

No declaration docstring.

theorem realClippedSection_eq_zero_of_T_lt
    (eta : ProgressiveL2Integrand filtration mu T)
    (truncationLevel : ℕ) (omega : Omega) {r : ℝ} (hr : (T : ℝ) < r) :
    realClippedSection eta truncationLevel omega r = 0 := by
  have hr0 : 0 ≤ r := T.property.trans hr.le
  rw [realClippedSection, realZeroExtension, if_pos hr0]
  rw [clippedExtensionAt, Function.extend_apply']
  · rfl
  · rintro ⟨s, hs⟩
    have hcoe := congrArg Prod.fst hs
    change (s.1 : ℝ≥0) = r.toNNReal at hcoe
    have hsT : (s.1 : ℝ) ≤ T := by exact_mod_cast s.1.property
    have hre : (r.toNNReal : ℝ) = r := by
      simp [Real.toNNReal_of_nonneg hr0]
    rw [← hre, ← hcoe] at hr
    exact (not_lt_of_ge hsT) hr

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:155published source at 7bcd37294df1

No declaration docstring.

theorem realClippedSection_abs_le
    (eta : ProgressiveL2Integrand filtration mu T)
    (truncationLevel : ℕ) (omega : Omega) (r : ℝ) :
    |realClippedSection eta truncationLevel omega r| ≤
      (truncationLevel : ℝ) := by
  by_cases hr : 0 ≤ r
  · rw [realClippedSection, realZeroExtension, if_pos hr]
    exact clippedExtensionAt_abs_le eta truncationLevel T r.toNNReal omega
  · rw [realClippedSection_eq_zero_of_neg eta truncationLevel omega
      (lt_of_not_ge hr)]
    simpa only [abs_zero] using
      (Nat.cast_nonneg truncationLevel : (0 : ℝ) ≤ (truncationLevel : ℝ))

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:172published source at 7bcd37294df1

No declaration docstring.

theorem realClippedSection_stronglyMeasurable
    (eta : ProgressiveL2Integrand filtration mu T)
    (truncationLevel : ℕ) (omega : Omega) :
    StronglyMeasurable (realClippedSection eta truncationLevel omega) := by
  apply realZeroExtension_stronglyMeasurable
  exact (clippedExtensionAt_stronglyMeasurable eta truncationLevel T).comp_measurable
    (measurable_id.prodMk measurable_const)

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:185published source at 7bcd37294df1

No declaration docstring.

theorem realClippedSection_integrable
    (eta : ProgressiveL2Integrand filtration mu T)
    (truncationLevel : ℕ) (omega : Omega) :
    Integrable (realClippedSection eta truncationLevel omega) volume := by
  have hon : IntegrableOn (realClippedSection eta truncationLevel omega)
      (Icc (0 : ℝ) (T : ℝ)) volume := by
    apply Measure.integrableOn_of_bounded (measure_Icc_lt_top.ne)
      (realClippedSection_stronglyMeasurable eta truncationLevel omega).aestronglyMeasurable
    filter_upwards [] with r
    simpa only [Real.norm_eq_abs] using
      realClippedSection_abs_le eta truncationLevel omega r
  have hind := hon.integrable_indicator measurableSet_Icc
  apply hind.congr
  filter_upwards [] with r
  by_cases hr : r ∈ Icc (0 : ℝ) (T : ℝ)
  · simp [hr]
  · have hout : r < 0 ∨ (T : ℝ) < r := by
      by_cases hr0 : 0 ≤ r
      · exact Or.inr (lt_of_not_ge (fun hrT ↦ hr ⟨hr0, hrT⟩))
      · exact Or.inl (lt_of_not_ge hr0)
    rcases hout with hneg | hTlt
    · rw [realClippedSection_eq_zero_of_neg eta truncationLevel omega hneg]
      simp [hr]
    · rw [realClippedSection_eq_zero_of_T_lt eta truncationLevel omega hTlt]
      simp [hr]

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:193published source at 7bcd37294df1

No declaration docstring.

theorem realClippedSection_locallyIntegrable
    (eta : ProgressiveL2Integrand filtration mu T)
    (truncationLevel : ℕ) (omega : Omega) :
    LocallyIntegrable (realClippedSection eta truncationLevel omega) volume :=
  (realClippedSection_integrable eta truncationLevel omega).locallyIntegrable

end TimeMeasureRealBridge
end StochasticProcesses
end TechnicalLemmas
end AutoSamplingTheory

AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:219published source at 7bcd37294df1