Theorem: MeasureTheory.WithDensityᵥEq.congr_ae
File: Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean

tactic: by_cases hf : Integrable f μ
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f✝ g✝ f g : α → E
h : f =ᶠ[ae μ] g
⊢ μ.withDensityᵥ f = μ.withDensityᵥ g
state_after: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f✝ g✝ f g : α → E
h : f =ᶠ[ae μ] g
hf : Integrable f μ
⊢ μ.withDensityᵥ f = μ.withDensityᵥ g

case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f✝ g✝ f g : α → E
h : f =ᶠ[ae μ] g
hf : ¬Integrable f μ
⊢ μ.withDensityᵥ f = μ.withDensityᵥ g

tactic: ext i hi
state_before: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f✝ g✝ f g : α → E
h : f =ᶠ[ae μ] g
hf : Integrable f μ
⊢ μ.withDensityᵥ f = μ.withDensityᵥ g
state_after: case pos.h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f✝ g✝ f g : α → E
h : f =ᶠ[ae μ] g
hf : Integrable f μ
i : Set α
hi : MeasurableSet i
⊢ ↑(μ.withDensityᵥ f) i = ↑(μ.withDensityᵥ g) i

tactic: rw [withDensityᵥ_apply hf hi, withDensityᵥ_apply (hf.congr h) hi]
state_before: case pos.h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f✝ g✝ f g : α → E
h : f =ᶠ[ae μ] g
hf : Integrable f μ
i : Set α
hi : MeasurableSet i
⊢ ↑(μ.withDensityᵥ f) i = ↑(μ.withDensityᵥ g) i
state_after: case pos.h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f✝ g✝ f g : α → E
h : f =ᶠ[ae μ] g
hf : Integrable f μ
i : Set α
hi : MeasurableSet i
⊢ ∫ (x : α) in i, f x ∂μ = ∫ (x : α) in i, g x ∂μ

tactic: exact integral_congr_ae (ae_restrict_of_ae h)
state_before: case pos.h
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f✝ g✝ f g : α → E
h : f =ᶠ[ae μ] g
hf : Integrable f μ
i : Set α
hi : MeasurableSet i
⊢ ∫ (x : α) in i, f x ∂μ = ∫ (x : α) in i, g x ∂μ
state_after: no goals

tactic: have hg : ¬Integrable g μ := by intro hg; exact hf (hg.congr h.symm)
state_before: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f✝ g✝ f g : α → E
h : f =ᶠ[ae μ] g
hf : ¬Integrable f μ
⊢ μ.withDensityᵥ f = μ.withDensityᵥ g
state_after: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f✝ g✝ f g : α → E
h : f =ᶠ[ae μ] g
hf : ¬Integrable f μ
hg : ¬Integrable g μ
⊢ μ.withDensityᵥ f = μ.withDensityᵥ g

tactic: rw [withDensityᵥ, withDensityᵥ, dif_neg hf, dif_neg hg]
state_before: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f✝ g✝ f g : α → E
h : f =ᶠ[ae μ] g
hf : ¬Integrable f μ
hg : ¬Integrable g μ
⊢ μ.withDensityᵥ f = μ.withDensityᵥ g
state_after: no goals

tactic: intro hg
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f✝ g✝ f g : α → E
h : f =ᶠ[ae μ] g
hf : ¬Integrable f μ
⊢ ¬Integrable g μ
state_after: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f✝ g✝ f g : α → E
h : f =ᶠ[ae μ] g
hf : ¬Integrable f μ
hg : Integrable g μ
⊢ False

tactic: exact hf (hg.congr h.symm)
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f✝ g✝ f g : α → E
h : f =ᶠ[ae μ] g
hf : ¬Integrable f μ
hg : Integrable g μ
⊢ False
state_after: no goals