Theorem: MeasureTheory.Integrable.prod_smul
File: Mathlib/MeasureTheory/Constructions/Prod/Integral.lean

tactic: refine (integrable_prod_iff ?_).2 ⟨?_, ?_⟩
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁸ : MeasurableSpace α
inst✝⁷ : MeasurableSpace α'
inst✝⁶ : MeasurableSpace β
inst✝⁵ : MeasurableSpace β'
inst✝⁴ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝³ : NormedAddCommGroup E
inst✝² : SigmaFinite ν
𝕜 : Type u_7
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedSpace 𝕜 E
f : α → 𝕜
g : β → E
hf : Integrable f μ
hg : Integrable g ν
⊢ Integrable (fun z => f z.1 • g z.2) (μ.prod ν)
state_after: case refine_1
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁸ : MeasurableSpace α
inst✝⁷ : MeasurableSpace α'
inst✝⁶ : MeasurableSpace β
inst✝⁵ : MeasurableSpace β'
inst✝⁴ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝³ : NormedAddCommGroup E
inst✝² : SigmaFinite ν
𝕜 : Type u_7
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedSpace 𝕜 E
f : α → 𝕜
g : β → E
hf : Integrable f μ
hg : Integrable g ν
⊢ AEStronglyMeasurable (fun z => f z.1 • g z.2) (μ.prod ν)

case refine_2
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁸ : MeasurableSpace α
inst✝⁷ : MeasurableSpace α'
inst✝⁶ : MeasurableSpace β
inst✝⁵ : MeasurableSpace β'
inst✝⁴ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝³ : NormedAddCommGroup E
inst✝² : SigmaFinite ν
𝕜 : Type u_7
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedSpace 𝕜 E
f : α → 𝕜
g : β → E
hf : Integrable f μ
hg : Integrable g ν
⊢ ∀ᵐ (x : α) ∂μ, Integrable (fun y => f (x, y).1 • g (x, y).2) ν

case refine_3
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁸ : MeasurableSpace α
inst✝⁷ : MeasurableSpace α'
inst✝⁶ : MeasurableSpace β
inst✝⁵ : MeasurableSpace β'
inst✝⁴ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝³ : NormedAddCommGroup E
inst✝² : SigmaFinite ν
𝕜 : Type u_7
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedSpace 𝕜 E
f : α → 𝕜
g : β → E
hf : Integrable f μ
hg : Integrable g ν
⊢ Integrable (fun x => ∫ (y : β), ‖f (x, y).1 • g (x, y).2‖ ∂ν) μ

tactic: exact hf.1.fst.smul hg.1.snd
state_before: case refine_1
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁸ : MeasurableSpace α
inst✝⁷ : MeasurableSpace α'
inst✝⁶ : MeasurableSpace β
inst✝⁵ : MeasurableSpace β'
inst✝⁴ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝³ : NormedAddCommGroup E
inst✝² : SigmaFinite ν
𝕜 : Type u_7
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedSpace 𝕜 E
f : α → 𝕜
g : β → E
hf : Integrable f μ
hg : Integrable g ν
⊢ AEStronglyMeasurable (fun z => f z.1 • g z.2) (μ.prod ν)
state_after: no goals

tactic: exact eventually_of_forall fun x => hg.smul (f x)
state_before: case refine_2
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁸ : MeasurableSpace α
inst✝⁷ : MeasurableSpace α'
inst✝⁶ : MeasurableSpace β
inst✝⁵ : MeasurableSpace β'
inst✝⁴ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝³ : NormedAddCommGroup E
inst✝² : SigmaFinite ν
𝕜 : Type u_7
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedSpace 𝕜 E
f : α → 𝕜
g : β → E
hf : Integrable f μ
hg : Integrable g ν
⊢ ∀ᵐ (x : α) ∂μ, Integrable (fun y => f (x, y).1 • g (x, y).2) ν
state_after: no goals

tactic: simpa only [norm_smul, integral_mul_left] using hf.norm.mul_const _
state_before: case refine_3
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁸ : MeasurableSpace α
inst✝⁷ : MeasurableSpace α'
inst✝⁶ : MeasurableSpace β
inst✝⁵ : MeasurableSpace β'
inst✝⁴ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝³ : NormedAddCommGroup E
inst✝² : SigmaFinite ν
𝕜 : Type u_7
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedSpace 𝕜 E
f : α → 𝕜
g : β → E
hf : Integrable f μ
hg : Integrable g ν
⊢ Integrable (fun x => ∫ (y : β), ‖f (x, y).1 • g (x, y).2‖ ∂ν) μ
state_after: no goals