Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
pp: case hm
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
⊢ ∫ (x : Ω), id ((PiLp.continuousLinearEquiv 2 ℝ fun a ↦ ℝ).symm (I.restrict fun x_1 ↦ X x_1 x)) ∂P =
    ∫ (x : ↥I → ℝ), id ((PiLp.continuousLinearEquiv 2 ℝ fun a ↦ ℝ).symm x) ∂projectiveFamily I

pp: case hm.hφ
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
⊢ AEMeasurable (⇑(PiLp.continuousLinearEquiv 2 ℝ fun a ↦ ℝ).symm) (projectiveFamily I)

pp: case hm.hfm
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
⊢ AEStronglyMeasurable id (Measure.map (⇑(PiLp.continuousLinearEquiv 2 ℝ fun a ↦ ℝ).symm) (projectiveFamily I))

pp: case hm.hφ
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
⊢ AEMeasurable (fun ω ↦ I.restrict fun x ↦ X x ω) P

pp: case hm.hfm
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
⊢ AEStronglyMeasurable (fun x ↦ id ((PiLp.continuousLinearEquiv 2 ℝ fun a ↦ ℝ).symm x))
    (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)

pp: case hm.hφ
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
⊢ AEMeasurable (⇑(PiLp.continuousLinearEquiv 2 ℝ fun a ↦ ℝ).symm) (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)

pp: case hm.hfm
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
⊢ AEStronglyMeasurable id
    (Measure.map (⇑(PiLp.continuousLinearEquiv 2 ℝ fun a ↦ ℝ).symm) (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P))

Tactic:
· simp only [id_eq]
        rw [ContinuousLinearEquiv.integral_comp_id_comm,
          ContinuousLinearEquiv.integral_comp_comm]
        simp only [PiLp.continuousLinearEquiv_symm_apply, integral_id_projectiveFamily,
          WithLp.toLp_zero, WithLp.toLp_eq_zero]
        congr with i
        rw [eval_integral]
        · simpa using h2 _
        · exact fun _ ↦ (h1.hasGaussianLaw_eval _).integrable

Goal state after:
case hm.hφ
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
⊢ AEMeasurable (⇑(PiLp.continuousLinearEquiv 2 ℝ fun a ↦ ℝ).symm) (projectiveFamily I)

case hm.hfm
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
⊢ AEStronglyMeasurable id (Measure.map (⇑(PiLp.continuousLinearEquiv 2 ℝ fun a ↦ ℝ).symm) (projectiveFamily I))

case hm.hφ
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
⊢ AEMeasurable (fun ω ↦ I.restrict fun x ↦ X x ω) P

case hm.hfm
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
⊢ AEStronglyMeasurable (fun x ↦ id ((PiLp.continuousLinearEquiv 2 ℝ fun a ↦ ℝ).symm x))
    (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)

case hm.hφ
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
⊢ AEMeasurable (⇑(PiLp.continuousLinearEquiv 2 ℝ fun a ↦ ℝ).symm) (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)

case hm.hfm
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
⊢ AEStronglyMeasurable id
    (Measure.map (⇑(PiLp.continuousLinearEquiv 2 ℝ fun a ↦ ℝ).symm) (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P))