l3lab/ntp-mathlib huggingface dataset:
theorem IntegrableOn.mul_continuousOn_of_subset (hg : IntegrableOn g A μ) (hg' : ContinuousOn g' K)
    (hA : MeasurableSet A) (hK : IsCompact K) (hAK : A ⊆ K) :
    IntegrableOn (fun x => g x * g' x) A μ := by
  rcases IsCompact.exists_bound_of_continuousOn hK hg' with ⟨C, hC⟩
  rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg ⊢
  

Proof state:
case intro
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f g✝ : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : Memℒp g 1
hg' : ContinuousOn g' K
hA : MeasurableSet A
hK : IsCompact K
hAK : A ⊆ K
C : ℝ
hC : ∀ x ∈ K, ‖g' x‖ ≤ C
⊢ Memℒp (fun x => g x * g' x) 1

Next tactic:
have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g x‖ := by
    filter_upwards [ae_restrict_mem hA] with x hx
    refine' (norm_mul_le _ _).trans _
    rw [mul_comm]
    apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)
theorem IntegrableOn.mul_continuousOn_of_subset (hg : IntegrableOn g A μ) (hg' : ContinuousOn g' K)
    (hA : MeasurableSet A) (hK : IsCompact K) (hAK : A ⊆ K) :
    IntegrableOn (fun x => g x * g' x) A μ := by
  rcases IsCompact.exists_bound_of_continuousOn hK hg' with ⟨C, hC⟩
  rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg ⊢
  have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g x‖ := by
    

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f g✝ : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : Memℒp g 1
hg' : ContinuousOn g' K
hA : MeasurableSet A
hK : IsCompact K
hAK : A ⊆ K
C : ℝ
hC : ∀ x ∈ K, ‖g' x‖ ≤ C
⊢ ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g x‖

Next tactic:
filter_upwards [ae_restrict_mem hA] with x hx
theorem IntegrableOn.mul_continuousOn_of_subset (hg : IntegrableOn g A μ) (hg' : ContinuousOn g' K)
    (hA : MeasurableSet A) (hK : IsCompact K) (hAK : A ⊆ K) :
    IntegrableOn (fun x => g x * g' x) A μ := by
  rcases IsCompact.exists_bound_of_continuousOn hK hg' with ⟨C, hC⟩
  rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg ⊢
  have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g x‖ := by
    filter_upwards [ae_restrict_mem hA] with x hx
    

Proof state:
case h
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f g✝ : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : Memℒp g 1
hg' : ContinuousOn g' K
hA : MeasurableSet A
hK : IsCompact K
hAK : A ⊆ K
C : ℝ
hC : ∀ x ∈ K, ‖g' x‖ ≤ C
x : X
hx : x ∈ A
⊢ ‖g x * g' x‖ ≤ C * ‖g x‖

Next tactic:
refine' (norm_mul_le _ _).trans _
theorem IntegrableOn.mul_continuousOn_of_subset (hg : IntegrableOn g A μ) (hg' : ContinuousOn g' K)
    (hA : MeasurableSet A) (hK : IsCompact K) (hAK : A ⊆ K) :
    IntegrableOn (fun x => g x * g' x) A μ := by
  rcases IsCompact.exists_bound_of_continuousOn hK hg' with ⟨C, hC⟩
  rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg ⊢
  have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g x‖ := by
    filter_upwards [ae_restrict_mem hA] with x hx
    refine' (norm_mul_le _ _).trans _
    

Proof state:
case h
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f g✝ : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : Memℒp g 1
hg' : ContinuousOn g' K
hA : MeasurableSet A
hK : IsCompact K
hAK : A ⊆ K
C : ℝ
hC : ∀ x ∈ K, ‖g' x‖ ≤ C
x : X
hx : x ∈ A
⊢ ‖g x‖ * ‖g' x‖ ≤ C * ‖g x‖

Next tactic:
rw [mul_comm]
theorem IntegrableOn.mul_continuousOn_of_subset (hg : IntegrableOn g A μ) (hg' : ContinuousOn g' K)
    (hA : MeasurableSet A) (hK : IsCompact K) (hAK : A ⊆ K) :
    IntegrableOn (fun x => g x * g' x) A μ := by
  rcases IsCompact.exists_bound_of_continuousOn hK hg' with ⟨C, hC⟩
  rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg ⊢
  have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g x‖ := by
    filter_upwards [ae_restrict_mem hA] with x hx
    refine' (norm_mul_le _ _).trans _
    rw [mul_comm]
    

Proof state:
case h
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f g✝ : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : Memℒp g 1
hg' : ContinuousOn g' K
hA : MeasurableSet A
hK : IsCompact K
hAK : A ⊆ K
C : ℝ
hC : ∀ x ∈ K, ‖g' x‖ ≤ C
x : X
hx : x ∈ A
⊢ ‖g' x‖ * ‖g x‖ ≤ C * ‖g x‖

Next tactic:
apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)
theorem IntegrableOn.mul_continuousOn_of_subset (hg : IntegrableOn g A μ) (hg' : ContinuousOn g' K)
    (hA : MeasurableSet A) (hK : IsCompact K) (hAK : A ⊆ K) :
    IntegrableOn (fun x => g x * g' x) A μ := by
  rcases IsCompact.exists_bound_of_continuousOn hK hg' with ⟨C, hC⟩
  rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg ⊢
  have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g x‖ := by
    filter_upwards [ae_restrict_mem hA] with x hx
    refine' (norm_mul_le _ _).trans _
    rw [mul_comm]
    apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)
  

Proof state:
case intro
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f g✝ : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : Memℒp g 1
hg' : ContinuousOn g' K
hA : MeasurableSet A
hK : IsCompact K
hAK : A ⊆ K
C : ℝ
hC : ∀ x ∈ K, ‖g' x‖ ≤ C
this : ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g x‖
⊢ Memℒp (fun x => g x * g' x) 1

Next tactic:
exact
    Memℒp.of_le_mul hg (hg.aestronglyMeasurable.mul <| (hg'.mono hAK).aestronglyMeasurable hA) this
theorem IntegrableOn.continuousOn_mul_of_subset (hg : ContinuousOn g K) (hg' : IntegrableOn g' A μ)
    (hK : IsCompact K) (hA : MeasurableSet A) (hAK : A ⊆ K) :
    IntegrableOn (fun x => g x * g' x) A μ := by
  

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f g✝ : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : ContinuousOn g K
hg' : IntegrableOn g' A
hK : IsCompact K
hA : MeasurableSet A
hAK : A ⊆ K
⊢ IntegrableOn (fun x => g x * g' x) A

Next tactic:
rcases IsCompact.exists_bound_of_continuousOn hK hg with ⟨C, hC⟩
theorem IntegrableOn.continuousOn_mul_of_subset (hg : ContinuousOn g K) (hg' : IntegrableOn g' A μ)
    (hK : IsCompact K) (hA : MeasurableSet A) (hAK : A ⊆ K) :
    IntegrableOn (fun x => g x * g' x) A μ := by
  rcases IsCompact.exists_bound_of_continuousOn hK hg with ⟨C, hC⟩
  

Proof state:
case intro
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f g✝ : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : ContinuousOn g K
hg' : IntegrableOn g' A
hK : IsCompact K
hA : MeasurableSet A
hAK : A ⊆ K
C : ℝ
hC : ∀ x ∈ K, ‖g x‖ ≤ C
⊢ IntegrableOn (fun x => g x * g' x) A

Next tactic:
rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg' ⊢
theorem IntegrableOn.continuousOn_mul_of_subset (hg : ContinuousOn g K) (hg' : IntegrableOn g' A μ)
    (hK : IsCompact K) (hA : MeasurableSet A) (hAK : A ⊆ K) :
    IntegrableOn (fun x => g x * g' x) A μ := by
  rcases IsCompact.exists_bound_of_continuousOn hK hg with ⟨C, hC⟩
  rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg' ⊢
  

Proof state:
case intro
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f g✝ : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : ContinuousOn g K
hg' : Memℒp g' 1
hK : IsCompact K
hA : MeasurableSet A
hAK : A ⊆ K
C : ℝ
hC : ∀ x ∈ K, ‖g x‖ ≤ C
⊢ Memℒp (fun x => g x * g' x) 1

Next tactic:
have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g' x‖ := by
    filter_upwards [ae_restrict_mem hA] with x hx
    refine' (norm_mul_le _ _).trans _
    apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)
theorem IntegrableOn.continuousOn_mul_of_subset (hg : ContinuousOn g K) (hg' : IntegrableOn g' A μ)
    (hK : IsCompact K) (hA : MeasurableSet A) (hAK : A ⊆ K) :
    IntegrableOn (fun x => g x * g' x) A μ := by
  rcases IsCompact.exists_bound_of_continuousOn hK hg with ⟨C, hC⟩
  rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg' ⊢
  have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g' x‖ := by
    

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f g✝ : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : ContinuousOn g K
hg' : Memℒp g' 1
hK : IsCompact K
hA : MeasurableSet A
hAK : A ⊆ K
C : ℝ
hC : ∀ x ∈ K, ‖g x‖ ≤ C
⊢ ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g' x‖

Next tactic:
filter_upwards [ae_restrict_mem hA] with x hx
theorem IntegrableOn.continuousOn_mul_of_subset (hg : ContinuousOn g K) (hg' : IntegrableOn g' A μ)
    (hK : IsCompact K) (hA : MeasurableSet A) (hAK : A ⊆ K) :
    IntegrableOn (fun x => g x * g' x) A μ := by
  rcases IsCompact.exists_bound_of_continuousOn hK hg with ⟨C, hC⟩
  rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg' ⊢
  have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g' x‖ := by
    filter_upwards [ae_restrict_mem hA] with x hx
    

Proof state:
case h
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f g✝ : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : ContinuousOn g K
hg' : Memℒp g' 1
hK : IsCompact K
hA : MeasurableSet A
hAK : A ⊆ K
C : ℝ
hC : ∀ x ∈ K, ‖g x‖ ≤ C
x : X
hx : x ∈ A
⊢ ‖g x * g' x‖ ≤ C * ‖g' x‖

Next tactic:
refine' (norm_mul_le _ _).trans _
theorem IntegrableOn.continuousOn_mul_of_subset (hg : ContinuousOn g K) (hg' : IntegrableOn g' A μ)
    (hK : IsCompact K) (hA : MeasurableSet A) (hAK : A ⊆ K) :
    IntegrableOn (fun x => g x * g' x) A μ := by
  rcases IsCompact.exists_bound_of_continuousOn hK hg with ⟨C, hC⟩
  rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg' ⊢
  have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g' x‖ := by
    filter_upwards [ae_restrict_mem hA] with x hx
    refine' (norm_mul_le _ _).trans _
    

Proof state:
case h
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f g✝ : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : ContinuousOn g K
hg' : Memℒp g' 1
hK : IsCompact K
hA : MeasurableSet A
hAK : A ⊆ K
C : ℝ
hC : ∀ x ∈ K, ‖g x‖ ≤ C
x : X
hx : x ∈ A
⊢ ‖g x‖ * ‖g' x‖ ≤ C * ‖g' x‖

Next tactic:
apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)
theorem IntegrableOn.continuousOn_mul_of_subset (hg : ContinuousOn g K) (hg' : IntegrableOn g' A μ)
    (hK : IsCompact K) (hA : MeasurableSet A) (hAK : A ⊆ K) :
    IntegrableOn (fun x => g x * g' x) A μ := by
  rcases IsCompact.exists_bound_of_continuousOn hK hg with ⟨C, hC⟩
  rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg' ⊢
  have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g' x‖ := by
    filter_upwards [ae_restrict_mem hA] with x hx
    refine' (norm_mul_le _ _).trans _
    apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)
  

Proof state:
case intro
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f g✝ : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : ContinuousOn g K
hg' : Memℒp g' 1
hK : IsCompact K
hA : MeasurableSet A
hAK : A ⊆ K
C : ℝ
hC : ∀ x ∈ K, ‖g x‖ ≤ C
this : ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g' x‖
⊢ Memℒp (fun x => g x * g' x) 1

Next tactic:
exact
    Memℒp.of_le_mul hg' (((hg.mono hAK).aestronglyMeasurable hA).mul hg'.aestronglyMeasurable) this
theorem IntegrableOn.continuousOn_smul [T2Space X] [SecondCountableTopologyEither X 𝕜] {g : X → E}
    (hg : IntegrableOn g K μ) {f : X → 𝕜} (hf : ContinuousOn f K) (hK : IsCompact K) :
    IntegrableOn (fun x => f x • g x) K μ := by
  

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
f✝ g✝ : X → E
μ : Measure X
s : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
𝕜 : Type u_6
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X 𝕜
g : X → E
hg : IntegrableOn g K
f : X → 𝕜
hf : ContinuousOn f K
hK : IsCompact K
⊢ IntegrableOn (fun x => f x • g x) K

Next tactic:
rw [IntegrableOn, ← integrable_norm_iff]
theorem IntegrableOn.continuousOn_smul [T2Space X] [SecondCountableTopologyEither X 𝕜] {g : X → E}
    (hg : IntegrableOn g K μ) {f : X → 𝕜} (hf : ContinuousOn f K) (hK : IsCompact K) :
    IntegrableOn (fun x => f x • g x) K μ := by
  rw [IntegrableOn, ← integrable_norm_iff]
  · 

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
f✝ g✝ : X → E
μ : Measure X
s : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
𝕜 : Type u_6
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X 𝕜
g : X → E
hg : IntegrableOn g K
f : X → 𝕜
hf : ContinuousOn f K
hK : IsCompact K
⊢ Integrable fun a => ‖f a • g a‖

Next tactic:
simp_rw [norm_smul]
theorem IntegrableOn.continuousOn_smul [T2Space X] [SecondCountableTopologyEither X 𝕜] {g : X → E}
    (hg : IntegrableOn g K μ) {f : X → 𝕜} (hf : ContinuousOn f K) (hK : IsCompact K) :
    IntegrableOn (fun x => f x • g x) K μ := by
  rw [IntegrableOn, ← integrable_norm_iff]
  · simp_rw [norm_smul]
    

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
f✝ g✝ : X → E
μ : Measure X
s : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
𝕜 : Type u_6
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X 𝕜
g : X → E
hg : IntegrableOn g K
f : X → 𝕜
hf : ContinuousOn f K
hK : IsCompact K
⊢ Integrable fun a => ‖f a‖ * ‖g a‖

Next tactic:
refine' IntegrableOn.continuousOn_mul _ hg.norm hK
theorem IntegrableOn.continuousOn_smul [T2Space X] [SecondCountableTopologyEither X 𝕜] {g : X → E}
    (hg : IntegrableOn g K μ) {f : X → 𝕜} (hf : ContinuousOn f K) (hK : IsCompact K) :
    IntegrableOn (fun x => f x • g x) K μ := by
  rw [IntegrableOn, ← integrable_norm_iff]
  · simp_rw [norm_smul]
    refine' IntegrableOn.continuousOn_mul _ hg.norm hK
    

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
f✝ g✝ : X → E
μ : Measure X
s : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
𝕜 : Type u_6
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X 𝕜
g : X → E
hg : IntegrableOn g K
f : X → 𝕜
hf : ContinuousOn f K
hK : IsCompact K
⊢ ContinuousOn (fun a => ‖f a‖) K

Next tactic:
exact continuous_norm.comp_continuousOn hf
theorem IntegrableOn.continuousOn_smul [T2Space X] [SecondCountableTopologyEither X 𝕜] {g : X → E}
    (hg : IntegrableOn g K μ) {f : X → 𝕜} (hf : ContinuousOn f K) (hK : IsCompact K) :
    IntegrableOn (fun x => f x • g x) K μ := by
  rw [IntegrableOn, ← integrable_norm_iff]
  · simp_rw [norm_smul]
    refine' IntegrableOn.continuousOn_mul _ hg.norm hK
    exact continuous_norm.comp_continuousOn hf
  · 

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
f✝ g✝ : X → E
μ : Measure X
s : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
𝕜 : Type u_6
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X 𝕜
g : X → E
hg : IntegrableOn g K
f : X → 𝕜
hf : ContinuousOn f K
hK : IsCompact K
⊢ AEStronglyMeasurable (fun x => f x • g x) (Measure.restrict μ K)

Next tactic:
exact (hf.aestronglyMeasurable hK.measurableSet).smul hg.1
theorem IntegrableOn.smul_continuousOn [T2Space X] [SecondCountableTopologyEither X E] {f : X → 𝕜}
    (hf : IntegrableOn f K μ) {g : X → E} (hg : ContinuousOn g K) (hK : IsCompact K) :
    IntegrableOn (fun x => f x • g x) K μ := by
  

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
f✝ g✝ : X → E
μ : Measure X
s : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
𝕜 : Type u_6
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X E
f : X → 𝕜
hf : IntegrableOn f K
g : X → E
hg : ContinuousOn g K
hK : IsCompact K
⊢ IntegrableOn (fun x => f x • g x) K

Next tactic:
rw [IntegrableOn, ← integrable_norm_iff]
theorem IntegrableOn.smul_continuousOn [T2Space X] [SecondCountableTopologyEither X E] {f : X → 𝕜}
    (hf : IntegrableOn f K μ) {g : X → E} (hg : ContinuousOn g K) (hK : IsCompact K) :
    IntegrableOn (fun x => f x • g x) K μ := by
  rw [IntegrableOn, ← integrable_norm_iff]
  · 

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
f✝ g✝ : X → E
μ : Measure X
s : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
𝕜 : Type u_6
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X E
f : X → 𝕜
hf : IntegrableOn f K
g : X → E
hg : ContinuousOn g K
hK : IsCompact K
⊢ Integrable fun a => ‖f a • g a‖

Next tactic:
simp_rw [norm_smul]
theorem IntegrableOn.smul_continuousOn [T2Space X] [SecondCountableTopologyEither X E] {f : X → 𝕜}
    (hf : IntegrableOn f K μ) {g : X → E} (hg : ContinuousOn g K) (hK : IsCompact K) :
    IntegrableOn (fun x => f x • g x) K μ := by
  rw [IntegrableOn, ← integrable_norm_iff]
  · simp_rw [norm_smul]
    

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
f✝ g✝ : X → E
μ : Measure X
s : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
𝕜 : Type u_6
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X E
f : X → 𝕜
hf : IntegrableOn f K
g : X → E
hg : ContinuousOn g K
hK : IsCompact K
⊢ Integrable fun a => ‖f a‖ * ‖g a‖

Next tactic:
refine' IntegrableOn.mul_continuousOn hf.norm _ hK
theorem IntegrableOn.smul_continuousOn [T2Space X] [SecondCountableTopologyEither X E] {f : X → 𝕜}
    (hf : IntegrableOn f K μ) {g : X → E} (hg : ContinuousOn g K) (hK : IsCompact K) :
    IntegrableOn (fun x => f x • g x) K μ := by
  rw [IntegrableOn, ← integrable_norm_iff]
  · simp_rw [norm_smul]
    refine' IntegrableOn.mul_continuousOn hf.norm _ hK
    

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
f✝ g✝ : X → E
μ : Measure X
s : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
𝕜 : Type u_6
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X E
f : X → 𝕜
hf : IntegrableOn f K
g : X → E
hg : ContinuousOn g K
hK : IsCompact K
⊢ ContinuousOn (fun a => ‖g a‖) K

Next tactic:
exact continuous_norm.comp_continuousOn hg
theorem IntegrableOn.smul_continuousOn [T2Space X] [SecondCountableTopologyEither X E] {f : X → 𝕜}
    (hf : IntegrableOn f K μ) {g : X → E} (hg : ContinuousOn g K) (hK : IsCompact K) :
    IntegrableOn (fun x => f x • g x) K μ := by
  rw [IntegrableOn, ← integrable_norm_iff]
  · simp_rw [norm_smul]
    refine' IntegrableOn.mul_continuousOn hf.norm _ hK
    exact continuous_norm.comp_continuousOn hg
  · 

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
f✝ g✝ : X → E
μ : Measure X
s : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
𝕜 : Type u_6
inst✝³ : NormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : T2Space X
inst✝ : SecondCountableTopologyEither X E
f : X → 𝕜
hf : IntegrableOn f K
g : X → E
hg : ContinuousOn g K
hK : IsCompact K
⊢ AEStronglyMeasurable (fun x => f x • g x) (Measure.restrict μ K)

Next tactic:
exact hf.1.smul (hg.aestronglyMeasurable hK.measurableSet)