-- Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
lemma _root_.MeasureTheory.AEStronglyMeasurable.fourierPowSMulRight
    (hf : AEStronglyMeasurable f μ) (n : ℕ) :
    AEStronglyMeasurable (fun v ↦ fourierPowSMulRight L f v n) μ

Proof state:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
μ : Measure V
inst✝ : SecondCountableTopology V
hf : AEStronglyMeasurable f μ
n : ℕ
⊢ AEStronglyMeasurable
    (fun v =>
      (-(2 * ↑π * I)) ^ n •
        ((smulRightL ℝ (fun x => W) E)
            ((ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight fun x => L v))
          (f v))
    μ

Tactic:
apply AEStronglyMeasurable.const_smul'

Result:
case hf
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
μ : Measure V
inst✝ : SecondCountableTopology V
hf : AEStronglyMeasurable f μ
n : ℕ
⊢ AEStronglyMeasurable
    (fun x =>
      ((smulRightL ℝ (fun x => W) E)
          ((ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight fun x_1 => L x))
        (f x))
    μ

Proof state:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
μ : Measure V
inst✝ : SecondCountableTopology V
hf : AEStronglyMeasurable f μ
n : ℕ
⊢ AEStronglyMeasurable (fun v => VectorFourier.fourierPowSMulRight L f v n) μ

Tactic:
simp_rw [fourierPowSMulRight_eq_comp]

Result:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
μ : Measure V
inst✝ : SecondCountableTopology V
hf : AEStronglyMeasurable f μ
n : ℕ
⊢ AEStronglyMeasurable
    (fun v =>
      (-(2 * ↑π * I)) ^ n •
        ((smulRightL ℝ (fun x => W) E)
            ((ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight fun x => L v))
          (f v))
    μ

Proof state:
case hf
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
μ : Measure V
inst✝ : SecondCountableTopology V
hf : AEStronglyMeasurable f μ
n : ℕ
⊢ Continuous fun x => (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight fun x_1 => L x

Tactic:
exact Continuous.comp (map_continuous _) (continuous_pi (fun _ ↦ L.continuous))

Result:
no goals

Proof state:
case hf
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
μ : Measure V
inst✝ : SecondCountableTopology V
hf : AEStronglyMeasurable f μ
n : ℕ
⊢ AEStronglyMeasurable
    (fun x =>
      ((smulRightL ℝ (fun x => W) E)
          ((ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight fun x_1 => L x))
        (f x))
    μ

Tactic:
apply (smulRightL ℝ (fun (_ : Fin n) ↦ W) E).continuous₂.comp_aestronglyMeasurable₂ _ hf

Result:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
μ : Measure V
inst✝ : SecondCountableTopology V
hf : AEStronglyMeasurable f μ
n : ℕ
⊢ AEStronglyMeasurable
    (fun x => (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight fun x_1 => L x) μ

Proof state:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
μ : Measure V
inst✝ : SecondCountableTopology V
hf : AEStronglyMeasurable f μ
n : ℕ
⊢ AEStronglyMeasurable
    (fun x => (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight fun x_1 => L x) μ

Tactic:
apply Continuous.aestronglyMeasurable

Result:
case hf
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : NormedSpace ℝ V
inst✝⁴ : NormedAddCommGroup W
inst✝³ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
μ : Measure V
inst✝ : SecondCountableTopology V
hf : AEStronglyMeasurable f μ
n : ℕ
⊢ Continuous fun x => (ContinuousMultilinearMap.mkPiAlgebra ℝ (Fin n) ℝ).compContinuousLinearMapLRight fun x_1 => L x