mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp_one : 1 ≤ p
hp_top : p ≠ ∞
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), MemLp (f i) p μ) → UnifIntegrable f p μ
f : Fin (n + 1) → α → β
hfLp : ∀ (i : Fin (n + 1)), MemLp (f i) p μ
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k ↦ f k.castSucc
hgLp : ∀ (i : Fin n), MemLp (g i) p μ
δ₁ : ℝ
hδ₁pos : 0 < δ₁
hδ₁ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₁ → eLpNorm (s.indicator (g i)) p μ ≤ ENNReal.ofReal ε
δ₂ : ℝ
hδ₂pos : 0 < δ₂
hδ₂ :
  ∀ (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₂ → eLpNorm (s.indicator (f (Fin.last n))) p μ ≤ ENNReal.ofReal ε
i : Fin (n + 1)
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
hi : ↑i < n
⊢ eLpNorm (s.indicator (g ⟨↑i, hi⟩)) p μ ≤ ENNReal.ofReal ε

pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp_one : 1 ≤ p
hp_top : p ≠ ∞
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), MemLp (f i) p μ) → UnifIntegrable f p μ
f : Fin (n + 1) → α → β
hfLp : ∀ (i : Fin (n + 1)), MemLp (f i) p μ
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k ↦ f k.castSucc
hgLp : ∀ (i : Fin n), MemLp (g i) p μ
δ₁ : ℝ
hδ₁pos : 0 < δ₁
hδ₁ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₁ → eLpNorm (s.indicator (g i)) p μ ≤ ENNReal.ofReal ε
δ₂ : ℝ
hδ₂pos : 0 < δ₂
hδ₂ :
  ∀ (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₂ → eLpNorm (s.indicator (f (Fin.last n))) p μ ≤ ENNReal.ofReal ε
i : Fin (n + 1)
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
hi : ↑i < n
⊢ f i = g ⟨↑i, hi⟩

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp_one : 1 ≤ p
hp_top : p ≠ ∞
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), MemLp (f i) p μ) → UnifIntegrable f p μ
f : Fin (n + 1) → α → β
hfLp : ∀ (i : Fin (n + 1)), MemLp (f i) p μ
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k ↦ f k.castSucc
hgLp : ∀ (i : Fin n), MemLp (g i) p μ
δ₁ : ℝ
hδ₁pos : 0 < δ₁
hδ₁ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₁ → eLpNorm (s.indicator (g i)) p μ ≤ ENNReal.ofReal ε
δ₂ : ℝ
hδ₂pos : 0 < δ₂
hδ₂ :
  ∀ (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₂ → eLpNorm (s.indicator (f (Fin.last n))) p μ ≤ ENNReal.ofReal ε
i : Fin (n + 1)
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
hi : ↑i < n
⊢ eLpNorm (s.indicator (g ⟨↑i, hi⟩)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : 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

pp: case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : 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

pp: case pos
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : 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

pp: case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : 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:
·

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : 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
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan

Goal state before:
pp: case h
α : Type u_1
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.toSignedMeasure = j₂.toSignedMeasure
S : Set α
hS₁ : MeasurableSet S
hS₂ : j₁.toSignedMeasure ≤[S] 0
hS₃ : 0 ≤[Sᶜ] j₁.toSignedMeasure
hS₄ : j₁.posPart S = 0
hS₅ : j₁.negPart Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : j₁.toSignedMeasure ≤[T] 0
hT₃ : 0 ≤[Tᶜ] j₁.toSignedMeasure
hT₄ : j₂.posPart T = 0
hT₅ : j₂.negPart Tᶜ = 0
hST₁ : j₁.toSignedMeasure (symmDiff Sᶜ Tᶜ) = 0
i : Set α
hi : MeasurableSet i
⊢ MeasurableSet (i ∩ Sᶜ)

Tactic:
·

Goal state after:
case h
α : Type u_1
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.toSignedMeasure = j₂.toSignedMeasure
S : Set α
hS₁ : MeasurableSet S
hS₂ : j₁.toSignedMeasure ≤[S] 0
hS₃ : 0 ≤[Sᶜ] j₁.toSignedMeasure
hS₄ : j₁.posPart S = 0
hS₅ : j₁.negPart Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : j₁.toSignedMeasure ≤[T] 0
hT₃ : 0 ≤[Tᶜ] j₁.toSignedMeasure
hT₄ : j₂.posPart T = 0
hT₅ : j₂.negPart Tᶜ = 0
hST₁ : j₁.toSignedMeasure (symmDiff Sᶜ Tᶜ) = 0
i : Set α
hi : MeasurableSet i
⊢ MeasurableSet (i ∩ Sᶜ)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : 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

pp: case pos
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : 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

pp: case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : 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:
·

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : 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
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : 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

pp: case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : 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:
·

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : 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
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : 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:
·

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
v : VectorMeasure α M
w : 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
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp_one : 1 ≤ p
hp_top : p ≠ ∞
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), MemLp (f i) p μ) → UnifIntegrable f p μ
f : Fin (n + 1) → α → β
hfLp : ∀ (i : Fin (n + 1)), MemLp (f i) p μ
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k ↦ f k.castSucc
hgLp : ∀ (i : Fin n), MemLp (g i) p μ
δ₁ : ℝ
hδ₁pos : 0 < δ₁
hδ₁ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₁ → eLpNorm (s.indicator (g i)) p μ ≤ ENNReal.ofReal ε
δ₂ : ℝ
hδ₂pos : 0 < δ₂
hδ₂ :
  ∀ (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₂ → eLpNorm (s.indicator (f (Fin.last n))) p μ ≤ ENNReal.ofReal ε
i : Fin (n + 1)
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
hi : ↑i < n
⊢ f i = g ⟨↑i, hi⟩

Tactic:
·

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp_one : 1 ≤ p
hp_top : p ≠ ∞
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), MemLp (f i) p μ) → UnifIntegrable f p μ
f : Fin (n + 1) → α → β
hfLp : ∀ (i : Fin (n + 1)), MemLp (f i) p μ
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k ↦ f k.castSucc
hgLp : ∀ (i : Fin n), MemLp (g i) p μ
δ₁ : ℝ
hδ₁pos : 0 < δ₁
hδ₁ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₁ → eLpNorm (s.indicator (g i)) p μ ≤ ENNReal.ofReal ε
δ₂ : ℝ
hδ₂pos : 0 < δ₂
hδ₂ :
  ∀ (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₂ → eLpNorm (s.indicator (f (Fin.last n))) p μ ≤ ENNReal.ofReal ε
i : Fin (n + 1)
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
hi : ↑i < n
⊢ f i = g ⟨↑i, hi⟩
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp_one : 1 ≤ p
hp_top : p ≠ ∞
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), MemLp (f i) p μ) → UnifIntegrable f p μ
f : Fin (n + 1) → α → β
hfLp : ∀ (i : Fin (n + 1)), MemLp (f i) p μ
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k ↦ f k.castSucc
hgLp : ∀ (i : Fin n), MemLp (g i) p μ
δ₁ : ℝ
hδ₁pos : 0 < δ₁
hδ₁ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₁ → eLpNorm (s.indicator (g i)) p μ ≤ ENNReal.ofReal ε
δ₂ : ℝ
hδ₂pos : 0 < δ₂
hδ₂ :
  ∀ (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₂ → eLpNorm (s.indicator (f (Fin.last n))) p μ ≤ ENNReal.ofReal ε
i : Fin (n + 1)
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
hi : n ≤ ↑i
⊢ eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp_one : 1 ≤ p
hp_top : p ≠ ∞
n : ℕ
h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), MemLp (f i) p μ) → UnifIntegrable f p μ
f : Fin (n + 1) → α → β
hfLp : ∀ (i : Fin (n + 1)), MemLp (f i) p μ
ε : ℝ
hε : 0 < ε
g : Fin n → α → β := fun k ↦ f k.castSucc
hgLp : ∀ (i : Fin n), MemLp (g i) p μ
δ₁ : ℝ
hδ₁pos : 0 < δ₁
hδ₁ :
  ∀ (i : Fin n) (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₁ → eLpNorm (s.indicator (g i)) p μ ≤ ENNReal.ofReal ε
δ₂ : ℝ
hδ₂pos : 0 < δ₂
hδ₂ :
  ∀ (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₂ → eLpNorm (s.indicator (f (Fin.last n))) p μ ≤ ENNReal.ofReal ε
i : Fin (n + 1)
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal (min δ₁ δ₂)
hi : n ≤ ↑i
⊢ eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case refine_1
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid N
inst✝² : TopologicalSpace N
v₁ v₂ : VectorMeasure α M
w : VectorMeasure α N
inst✝¹ : T2Space N
inst✝ : ContinuousAdd M
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ t ⊆ u, v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, v₂ t = 0
hv₂ : ∀ t ⊆ vᶜ, w t = 0
t : Set α
ht : t ⊆ u ∩ v
x✝ : MeasurableSet t
⊢ (v₁ + v₂) t = 0

pp: case refine_2
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid N
inst✝² : TopologicalSpace N
v₁ v₂ : VectorMeasure α M
w : VectorMeasure α N
inst✝¹ : T2Space N
inst✝ : ContinuousAdd M
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ t ⊆ u, v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, v₂ t = 0
hv₂ : ∀ t ⊆ vᶜ, w t = 0
t : Set α
ht : t ⊆ (u ∩ v)ᶜ
hmt : MeasurableSet t
⊢ w t = 0

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid N
inst✝² : TopologicalSpace N
v₁ v₂ : VectorMeasure α M
w : VectorMeasure α N
inst✝¹ : T2Space N
inst✝ : ContinuousAdd M
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ t ⊆ u, v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, v₂ t = 0
hv₂ : ∀ t ⊆ vᶜ, w t = 0
t : Set α
ht : t ⊆ u ∩ v
x✝ : MeasurableSet t
⊢ (v₁ + v₂) t = 0
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case refine_2
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid N
inst✝² : TopologicalSpace N
v₁ v₂ : VectorMeasure α M
w : VectorMeasure α N
inst✝¹ : T2Space N
inst✝ : ContinuousAdd M
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ t ⊆ u, v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, v₂ t = 0
hv₂ : ∀ t ⊆ vᶜ, w t = 0
t : Set α
ht : t ⊆ (u ∩ v)ᶜ
hmt : MeasurableSet t
⊢ w t = 0

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid N
inst✝² : TopologicalSpace N
v₁ v₂ : VectorMeasure α M
w : VectorMeasure α N
inst✝¹ : T2Space N
inst✝ : ContinuousAdd M
u : Set α
hmu : MeasurableSet u
hu₁ : ∀ t ⊆ u, v₁ t = 0
hu₂ : ∀ t ⊆ uᶜ, w t = 0
v : Set α
hmv : MeasurableSet v
hv₁ : ∀ t ⊆ v, v₂ t = 0
hv₂ : ∀ t ⊆ vᶜ, w t = 0
t : Set α
ht : t ⊆ (u ∩ v)ᶜ
hmt : MeasurableSet t
⊢ w t = 0
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ∞ ≤ ε
⊢ ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p μ ≤ ε

pp: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ∞
⊢ ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p μ ≤ ε

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ∞ ≤ ε
⊢ ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p μ ≤ ε
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan

Goal state before:
pp: case hd
α : Type u_1
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.toSignedMeasure = j₂.toSignedMeasure
S : Set α
hS₁ : MeasurableSet S
hS₂ : j₁.toSignedMeasure ≤[S] 0
hS₃ : 0 ≤[Sᶜ] j₁.toSignedMeasure
hS₄ : j₁.posPart S = 0
hS₅ : j₁.negPart Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : j₁.toSignedMeasure ≤[T] 0
hT₃ : 0 ≤[Tᶜ] j₁.toSignedMeasure
hT₄ : j₂.posPart T = 0
hT₅ : j₂.negPart Tᶜ = 0
hST₁ : j₁.toSignedMeasure (symmDiff Sᶜ Tᶜ) = 0
i : Set α
hi : MeasurableSet i
hμ₁ : j₁.posPart.real i = j₁.toSignedMeasure (i ∩ Sᶜ)
⊢ Disjoint (i ∩ T) (i ∩ Tᶜ)

pp: case h
α : Type u_1
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.toSignedMeasure = j₂.toSignedMeasure
S : Set α
hS₁ : MeasurableSet S
hS₂ : j₁.toSignedMeasure ≤[S] 0
hS₃ : 0 ≤[Sᶜ] j₁.toSignedMeasure
hS₄ : j₁.posPart S = 0
hS₅ : j₁.negPart Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : j₁.toSignedMeasure ≤[T] 0
hT₃ : 0 ≤[Tᶜ] j₁.toSignedMeasure
hT₄ : j₂.posPart T = 0
hT₅ : j₂.negPart Tᶜ = 0
hST₁ : j₁.toSignedMeasure (symmDiff Sᶜ Tᶜ) = 0
i : Set α
hi : MeasurableSet i
hμ₁ : j₁.posPart.real i = j₁.toSignedMeasure (i ∩ Sᶜ)
⊢ MeasurableSet (i ∩ Tᶜ)

Tactic:
·

Goal state after:
case hd
α : Type u_1
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.toSignedMeasure = j₂.toSignedMeasure
S : Set α
hS₁ : MeasurableSet S
hS₂ : j₁.toSignedMeasure ≤[S] 0
hS₃ : 0 ≤[Sᶜ] j₁.toSignedMeasure
hS₄ : j₁.posPart S = 0
hS₅ : j₁.negPart Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : j₁.toSignedMeasure ≤[T] 0
hT₃ : 0 ≤[Tᶜ] j₁.toSignedMeasure
hT₄ : j₂.posPart T = 0
hT₅ : j₂.negPart Tᶜ = 0
hST₁ : j₁.toSignedMeasure (symmDiff Sᶜ Tᶜ) = 0
i : Set α
hi : MeasurableSet i
hμ₁ : j₁.posPart.real i = j₁.toSignedMeasure (i ∩ Sᶜ)
⊢ Disjoint (i ∩ T) (i ∩ Tᶜ)
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan

Goal state before:
pp: case h
α : Type u_1
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.toSignedMeasure = j₂.toSignedMeasure
S : Set α
hS₁ : MeasurableSet S
hS₂ : j₁.toSignedMeasure ≤[S] 0
hS₃ : 0 ≤[Sᶜ] j₁.toSignedMeasure
hS₄ : j₁.posPart S = 0
hS₅ : j₁.negPart Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : j₁.toSignedMeasure ≤[T] 0
hT₃ : 0 ≤[Tᶜ] j₁.toSignedMeasure
hT₄ : j₂.posPart T = 0
hT₅ : j₂.negPart Tᶜ = 0
hST₁ : j₁.toSignedMeasure (symmDiff Sᶜ Tᶜ) = 0
i : Set α
hi : MeasurableSet i
hμ₁ : j₁.posPart.real i = j₁.toSignedMeasure (i ∩ Sᶜ)
⊢ MeasurableSet (i ∩ Tᶜ)

Tactic:
·

Goal state after:
case h
α : Type u_1
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.toSignedMeasure = j₂.toSignedMeasure
S : Set α
hS₁ : MeasurableSet S
hS₂ : j₁.toSignedMeasure ≤[S] 0
hS₃ : 0 ≤[Sᶜ] j₁.toSignedMeasure
hS₄ : j₁.posPart S = 0
hS₅ : j₁.negPart Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : j₁.toSignedMeasure ≤[T] 0
hT₃ : 0 ≤[Tᶜ] j₁.toSignedMeasure
hT₄ : j₂.posPart T = 0
hT₅ : j₂.negPart Tᶜ = 0
hST₁ : j₁.toSignedMeasure (symmDiff Sᶜ Tᶜ) = 0
i : Set α
hi : MeasurableSet i
hμ₁ : j₁.posPart.real i = j₁.toSignedMeasure (i ∩ Sᶜ)
⊢ MeasurableSet (i ∩ Tᶜ)
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ∞
hμ : μ = 0
⊢ ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p μ ≤ ε

pp: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ∞
hμ : ¬μ = 0
⊢ ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p μ ≤ ε

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
h : ε < ∞
hμ : μ = 0
⊢ ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p μ ≤ ε
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan

Goal state before:
pp: case pos
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
r : ℝ
hr : 0 ≤ r
⊢ (r • s).toJordanDecomposition = r • s.toJordanDecomposition

pp: case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
r : ℝ
hr : r < 0
⊢ (r • s).toJordanDecomposition = r • s.toJordanDecomposition

Tactic:
·

Goal state after:
case pos
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
r : ℝ
hr : 0 ≤ r
⊢ (r • s).toJordanDecomposition = r • s.toJordanDecomposition
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan

Goal state before:
pp: case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
r : ℝ
hr : r < 0
⊢ (r • s).toJordanDecomposition = r • s.toJordanDecomposition

Tactic:
·

Goal state after:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
r : ℝ
hr : r < 0
⊢ (r • s).toJordanDecomposition = r • s.toJordanDecomposition
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan

Goal state before:
pp: case neg.posPart
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
r : ℝ
hr : r < 0
⊢ (r • s).toJordanDecomposition.posPart = (r • s.toJordanDecomposition).posPart

pp: case neg.negPart
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
r : ℝ
hr : r < 0
⊢ (r • s).toJordanDecomposition.negPart = (r • s.toJordanDecomposition).negPart

Tactic:
·

Goal state after:
case neg.posPart
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
r : ℝ
hr : r < 0
⊢ (r • s).toJordanDecomposition.posPart = (r • s.toJordanDecomposition).posPart
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan

Goal state before:
pp: case neg.negPart
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
r : ℝ
hr : r < 0
⊢ (r • s).toJordanDecomposition.negPart = (r • s.toJordanDecomposition).negPart

Tactic:
·

Goal state after:
case neg.negPart
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
r : ℝ
hr : r < 0
⊢ (r • s).toJordanDecomposition.negPart = (r • s.toJordanDecomposition).negPart