Theorem: MeasureTheory.VectorMeasure.MutuallySingular.mk
File: Mathlib/MeasureTheory/Measure/VectorMeasure.lean

tactic: refine ⟨s, hs, fun t hst => ?_, fun t hst => ?_⟩ <;> by_cases ht : MeasurableSet t
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
s : Set α
hs : MeasurableSet s
h₁ : ∀ t ⊆ s, MeasurableSet t → ↑v t = 0
h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → ↑w t = 0
⊢ v ⟂ᵥ w
state_after: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
s : Set α
hs : MeasurableSet s
h₁ : ∀ t ⊆ s, MeasurableSet t → ↑v t = 0
h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → ↑w t = 0
t : Set α
hst : t ⊆ s
ht : MeasurableSet t
⊢ ↑v t = 0

case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
s : Set α
hs : MeasurableSet s
h₁ : ∀ t ⊆ s, MeasurableSet t → ↑v t = 0
h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → ↑w t = 0
t : Set α
hst : t ⊆ s
ht : ¬MeasurableSet t
⊢ ↑v t = 0

case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
s : Set α
hs : MeasurableSet s
h₁ : ∀ t ⊆ s, MeasurableSet t → ↑v t = 0
h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → ↑w t = 0
t : Set α
hst : t ⊆ sᶜ
ht : MeasurableSet t
⊢ ↑w t = 0

case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
s : Set α
hs : MeasurableSet s
h₁ : ∀ t ⊆ s, MeasurableSet t → ↑v t = 0
h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → ↑w t = 0
t : Set α
hst : t ⊆ sᶜ
ht : ¬MeasurableSet t
⊢ ↑w t = 0

tactic: exact h₁ t hst ht
state_before: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
s : Set α
hs : MeasurableSet s
h₁ : ∀ t ⊆ s, MeasurableSet t → ↑v t = 0
h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → ↑w t = 0
t : Set α
hst : t ⊆ s
ht : MeasurableSet t
⊢ ↑v t = 0
state_after: no goals

tactic: exact not_measurable v ht
state_before: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
s : Set α
hs : MeasurableSet s
h₁ : ∀ t ⊆ s, MeasurableSet t → ↑v t = 0
h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → ↑w t = 0
t : Set α
hst : t ⊆ s
ht : ¬MeasurableSet t
⊢ ↑v t = 0
state_after: no goals

tactic: exact h₂ t hst ht
state_before: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
s : Set α
hs : MeasurableSet s
h₁ : ∀ t ⊆ s, MeasurableSet t → ↑v t = 0
h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → ↑w t = 0
t : Set α
hst : t ⊆ sᶜ
ht : MeasurableSet t
⊢ ↑w t = 0
state_after: no goals

tactic: exact not_measurable w ht
state_before: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v v₁ v₂ : VectorMeasure α M
w w₁ w₂ : VectorMeasure α N
s : Set α
hs : MeasurableSet s
h₁ : ∀ t ⊆ s, MeasurableSet t → ↑v t = 0
h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → ↑w t = 0
t : Set α
hst : t ⊆ sᶜ
ht : ¬MeasurableSet t
⊢ ↑w t = 0
state_after: no goals