case intro
X : Type u_1
Y : Type ?u.2480793
E : Type ?u.2480796
R : Type u_2
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : 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 : X), x ∈ K → ‖g x‖ ≤ C
⊢ IntegrableOn (fun x => g x * g' x) A
X : Type u_1
Y : Type ?u.2480793
E : Type ?u.2480796
R : Type u_2
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : 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
rcases IsCompact.exists_bound_of_continuousOn hK hg with ⟨C, hC⟩
case intro
X : Type u_1
Y : Type ?u.2480793
E : Type ?u.2480796
R : Type u_2
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : 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 : X), x ∈ K → ‖g x‖ ≤ C
⊢ Memℒp (fun x => g x * g' x) 1
case intro
X : Type u_1
Y : Type ?u.2480793
E : Type ?u.2480796
R : Type u_2
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : 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 : X), x ∈ K → ‖g x‖ ≤ C
⊢ IntegrableOn (fun x => g x * g' x) A
rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg'⊢
case intro
X : Type u_1
Y : Type ?u.2480793
E : Type ?u.2480796
R : Type u_2
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : 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 : X), 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
case intro
X : Type u_1
Y : Type ?u.2480793
E : Type ?u.2480796
R : Type u_2
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : 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 : X), x ∈ K → ‖g x‖ ≤ C
⊢ Memℒp (fun x => g x * g' x) 1
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 _)
no goals
case intro
X : Type u_1
Y : Type ?u.2480793
E : Type ?u.2480796
R : Type u_2
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : 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 : X), 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
exact
  Memℒp.of_le_mul hg' (((hg.mono hAK).aestronglyMeasurable hA).mul hg'.aestronglyMeasurable) this
case h
X : Type u_1
Y : Type ?u.2480793
E : Type ?u.2480796
R : Type u_2
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : 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 : X), x ∈ K → ‖g x‖ ≤ C
x : X
hx : x ∈ A
⊢ ‖g x * g' x‖ ≤ C * ‖g' x‖
X : Type u_1
Y : Type ?u.2480793
E : Type ?u.2480796
R : Type u_2
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : 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 : X), x ∈ K → ‖g x‖ ≤ C
⊢ ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g' x‖
filter_upwards [ae_restrict_mem hA]with x hx
case h
X : Type u_1
Y : Type ?u.2480793
E : Type ?u.2480796
R : Type u_2
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : 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 : X), x ∈ K → ‖g x‖ ≤ C
x : X
hx : x ∈ A
⊢ ‖g x‖ * ‖g' x‖ ≤ C * ‖g' x‖
case h
X : Type u_1
Y : Type ?u.2480793
E : Type ?u.2480796
R : Type u_2
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : 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 : X), x ∈ K → ‖g x‖ ≤ C
x : X
hx : x ∈ A
⊢ ‖g x * g' x‖ ≤ C * ‖g' x‖
refine' (norm_mul_le _ _).trans _
no goals
case h
X : Type u_1
Y : Type ?u.2480793
E : Type ?u.2480796
R : Type u_2
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace Y
inst✝⁴ : TopologicalSpace Y
inst✝³ : NormedAddCommGroup E
f : 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 : X), x ∈ K → ‖g x‖ ≤ C
x : X
hx : x ∈ A
⊢ ‖g x‖ * ‖g' x‖ ≤ C * ‖g' x‖
apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)