Theorem: MeasureTheory.Measure.withDensityᵥ_absolutelyContinuous
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 : α → E
μ : Measure α
f : α → ℝ
⊢ μ.withDensityᵥ f ≪ᵥ μ.toENNRealVectorMeasure
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 : α → E
μ : Measure α
f : α → ℝ
hf : Integrable f μ
⊢ μ.withDensityᵥ f ≪ᵥ μ.toENNRealVectorMeasure

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 : α → E
μ : Measure α
f : α → ℝ
hf : ¬Integrable f μ
⊢ μ.withDensityᵥ f ≪ᵥ μ.toENNRealVectorMeasure

tactic: refine VectorMeasure.AbsolutelyContinuous.mk fun i hi₁ hi₂ => ?_
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 : α → E
μ : Measure α
f : α → ℝ
hf : Integrable f μ
⊢ μ.withDensityᵥ f ≪ᵥ μ.toENNRealVectorMeasure
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 : α → E
μ : Measure α
f : α → ℝ
hf : Integrable f μ
i : Set α
hi₁ : MeasurableSet i
hi₂ : ↑μ.toENNRealVectorMeasure i = 0
⊢ ↑(μ.withDensityᵥ f) i = 0

tactic: rw [toENNRealVectorMeasure_apply_measurable hi₁] at hi₂
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 : α → E
μ : Measure α
f : α → ℝ
hf : Integrable f μ
i : Set α
hi₁ : MeasurableSet i
hi₂ : ↑μ.toENNRealVectorMeasure i = 0
⊢ ↑(μ.withDensityᵥ f) i = 0
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 : α → E
μ : Measure α
f : α → ℝ
hf : Integrable f μ
i : Set α
hi₁ : MeasurableSet i
hi₂ : μ i = 0
⊢ ↑(μ.withDensityᵥ f) i = 0

tactic: rw [withDensityᵥ_apply hf hi₁, Measure.restrict_zero_set hi₂, integral_zero_measure]
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 : α → E
μ : Measure α
f : α → ℝ
hf : Integrable f μ
i : Set α
hi₁ : MeasurableSet i
hi₂ : μ i = 0
⊢ ↑(μ.withDensityᵥ f) i = 0
state_after: no goals

tactic: rw [withDensityᵥ, dif_neg hf]
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 : α → E
μ : Measure α
f : α → ℝ
hf : ¬Integrable f μ
⊢ μ.withDensityᵥ f ≪ᵥ μ.toENNRealVectorMeasure
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 : α → E
μ : Measure α
f : α → ℝ
hf : ¬Integrable f μ
⊢ 0 ≪ᵥ μ.toENNRealVectorMeasure

tactic: exact VectorMeasure.AbsolutelyContinuous.zero _
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 : α → E
μ : Measure α
f : α → ℝ
hf : ¬Integrable f μ
⊢ 0 ≪ᵥ μ.toENNRealVectorMeasure
state_after: no goals