mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn

Goal state before:
pp: case e'_4
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
A : Set α := i \ ⋃ l, s.restrictNonposSeq i l
hA : A = i \ ⋃ l, s.restrictNonposSeq i l
bdd : ℕ → ℕ := fun n ↦ s.findExistsOneDivLT (i \ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k)
n : ℕ
⊢ (fun l ↦ ⋃ (_ : l ≤ n), s.restrictNonposSeq i l) = fun l ↦ ⋃ (_ : l < n + 1), s.restrictNonposSeq i l

Tactic:
·

Goal state after:
case e'_4
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
A : Set α := i \ ⋃ l, s.restrictNonposSeq i l
hA : A = i \ ⋃ l, s.restrictNonposSeq i l
bdd : ℕ → ℕ := fun n ↦ s.findExistsOneDivLT (i \ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k)
n : ℕ
⊢ (fun l ↦ ⋃ (_ : l ≤ n), s.restrictNonposSeq i l) = fun l ↦ ⋃ (_ : l < n + 1), s.restrictNonposSeq i l
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v : VectorMeasure α M
i : Set α
hi₂ : v ≤[i] 0
hi₁ : ¬MeasurableSet i
⊢ v i ≤ 0

Tactic:
·

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid M
inst✝ : PartialOrder M
v : VectorMeasure α M
i : Set α
hi₂ : v ≤[i] 0
hi₁ : ¬MeasurableSet i
⊢ v i ≤ 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
h : ∀ ⦃s : Set α⦄, MeasurableSet s → w s = 0 → v s = 0
s : Set α
hs : w s = 0
hmeas : MeasurableSet s
⊢ v s = 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
h : ∀ ⦃s : Set α⦄, MeasurableSet s → w s = 0 → v s = 0
s : Set α
hs : w s = 0
hmeas : ¬MeasurableSet s
⊢ v s = 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
h : ∀ ⦃s : Set α⦄, MeasurableSet s → w s = 0 → v s = 0
s : Set α
hs : w s = 0
hmeas : MeasurableSet s
⊢ v s = 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
h : ∀ ⦃s : Set α⦄, MeasurableSet s → w s = 0 → v s = 0
s : Set α
hs : w s = 0
hmeas : ¬MeasurableSet s
⊢ v s = 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
h : ∀ ⦃s : Set α⦄, MeasurableSet s → w s = 0 → v s = 0
s : Set α
hs : w s = 0
hmeas : ¬MeasurableSet s
⊢ v s = 0
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn

Goal state before:
pp: case hA
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
A : Set α := i \ ⋃ l, s.restrictNonposSeq i l
hA : A = i \ ⋃ l, s.restrictNonposSeq i l
bdd : ℕ → ℕ := fun n ↦ s.findExistsOneDivLT (i \ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k)
hn' : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l ≤ n), s.restrictNonposSeq i l] 0
⊢ MeasurableSet (⋃ i_1, s.restrictNonposSeq i i_1)

pp: case hB
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
A : Set α := i \ ⋃ l, s.restrictNonposSeq i l
hA : A = i \ ⋃ l, s.restrictNonposSeq i l
bdd : ℕ → ℕ := fun n ↦ s.findExistsOneDivLT (i \ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k)
hn' : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l ≤ n), s.restrictNonposSeq i l] 0
⊢ MeasurableSet i

pp: case h
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
A : Set α := i \ ⋃ l, s.restrictNonposSeq i l
hA : A = i \ ⋃ l, s.restrictNonposSeq i l
bdd : ℕ → ℕ := fun n ↦ s.findExistsOneDivLT (i \ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k)
hn' : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l ≤ n), s.restrictNonposSeq i l] 0
⊢ ⋃ i_1, s.restrictNonposSeq i i_1 ⊆ i

pp: case hm
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
A : Set α := i \ ⋃ l, s.restrictNonposSeq i l
hA : A = i \ ⋃ l, s.restrictNonposSeq i l
bdd : ℕ → ℕ := fun n ↦ s.findExistsOneDivLT (i \ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k)
hn' : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l ≤ n), s.restrictNonposSeq i l] 0
⊢ ∀ (i_1 : ℕ), MeasurableSet (s.restrictNonposSeq i i_1)

pp: case hd
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
A : Set α := i \ ⋃ l, s.restrictNonposSeq i l
hA : A = i \ ⋃ l, s.restrictNonposSeq i l
bdd : ℕ → ℕ := fun n ↦ s.findExistsOneDivLT (i \ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k)
hn' : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l ≤ n), s.restrictNonposSeq i l] 0
⊢ Pairwise (Function.onFun Disjoint (s.restrictNonposSeq i))

Tactic:
·

Goal state after:
case hA
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
A : Set α := i \ ⋃ l, s.restrictNonposSeq i l
hA : A = i \ ⋃ l, s.restrictNonposSeq i l
bdd : ℕ → ℕ := fun n ↦ s.findExistsOneDivLT (i \ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k)
hn' : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l ≤ n), s.restrictNonposSeq i l] 0
⊢ MeasurableSet (⋃ i_1, s.restrictNonposSeq i i_1)
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : Subsingleton ι
hp_one : 1 ≤ p
hp_top : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), MemLp (f i) p μ
ε : ℝ
hε : 0 < ε
hι : Nonempty ι
⊢ ∃ δ,
    ∃ (_ : 0 < δ),
      ∀ (i : ι) (s : Set α),
        MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε

pp: case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : Subsingleton ι
hp_one : 1 ≤ p
hp_top : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), MemLp (f i) p μ
ε : ℝ
hε : 0 < ε
hι : ¬Nonempty ι
⊢ ∃ δ,
    ∃ (_ : 0 < δ),
      ∀ (i : ι) (s : Set α),
        MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : Subsingleton ι
hp_one : 1 ≤ p
hp_top : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), MemLp (f i) p μ
ε : ℝ
hε : 0 < ε
hι : Nonempty ι
⊢ ∃ δ,
    ∃ (_ : 0 < δ),
      ∀ (i : ι) (s : Set α),
        MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝⁴ : AddCommMonoid M
inst✝³ : TopologicalSpace M
inst✝² : AddCommMonoid N
inst✝¹ : TopologicalSpace N
v : VectorMeasure α M
w : VectorMeasure α N
inst✝ : MeasureSpace β
h : v ≪ᵥ w
f : α → β
hf : Measurable f
⊢ v.map f ≪ᵥ w.map f

pp: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝⁴ : AddCommMonoid M
inst✝³ : TopologicalSpace M
inst✝² : AddCommMonoid N
inst✝¹ : TopologicalSpace N
v : VectorMeasure α M
w : VectorMeasure α N
inst✝ : MeasureSpace β
h : v ≪ᵥ w
f : α → β
hf : ¬Measurable f
⊢ v.map f ≪ᵥ w.map f

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝⁴ : AddCommMonoid M
inst✝³ : TopologicalSpace M
inst✝² : AddCommMonoid N
inst✝¹ : TopologicalSpace N
v : VectorMeasure α M
w : VectorMeasure α N
inst✝ : MeasureSpace β
h : v ≪ᵥ w
f : α → β
hf : Measurable f
⊢ v.map f ≪ᵥ w.map f
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : Subsingleton ι
hp_one : 1 ≤ p
hp_top : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), MemLp (f i) p μ
ε : ℝ
hε : 0 < ε
hι : ¬Nonempty ι
⊢ ∃ δ,
    ∃ (_ : 0 < δ),
      ∀ (i : ι) (s : Set α),
        MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : Subsingleton ι
hp_one : 1 ≤ p
hp_top : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), MemLp (f i) p μ
ε : ℝ
hε : 0 < ε
hι : ¬Nonempty ι
⊢ ∃ δ,
    ∃ (_ : 0 < δ),
      ∀ (i : ι) (s : Set α),
        MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝⁴ : AddCommMonoid M
inst✝³ : TopologicalSpace M
inst✝² : AddCommMonoid N
inst✝¹ : TopologicalSpace N
v : VectorMeasure α M
w : VectorMeasure α N
inst✝ : MeasureSpace β
h : v ≪ᵥ w
f : α → β
hf : ¬Measurable f
⊢ v.map f ≪ᵥ w.map f

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝⁴ : AddCommMonoid M
inst✝³ : TopologicalSpace M
inst✝² : AddCommMonoid N
inst✝¹ : TopologicalSpace N
v : VectorMeasure α M
w : VectorMeasure α N
inst✝ : MeasureSpace β
h : v ≪ᵥ w
f : α → β
hf : ¬Measurable f
⊢ v.map f ≪ᵥ w.map f
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝⁴ : AddCommMonoid M
inst✝³ : TopologicalSpace M
inst✝² : AddCommMonoid N
inst✝¹ : TopologicalSpace N
v : VectorMeasure α M
w : VectorMeasure α N
inst✝ : MeasureSpace β
h : v ≪ᵥ w
f : α → β
hf : ¬Measurable f
s : Set β
a✝ : (w.map f) s = 0
⊢ (v.map f) s = 0

Tactic:
map_not_measurable v hf,

Goal state after:
case neg
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝⁴ : AddCommMonoid M
inst✝³ : TopologicalSpace M
inst✝² : AddCommMonoid N
inst✝¹ : TopologicalSpace N
v : VectorMeasure α M
w : VectorMeasure α N
inst✝ : MeasureSpace β
h : v ≪ᵥ w
f : α → β
hf : ¬Measurable f
s : Set β
a✝ : (w.map f) s = 0
⊢ 0 s = 0
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case mp
α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v : VectorMeasure α M
μ : VectorMeasure α ℝ≥0∞
h : ∀ ⦃s : Set α⦄, μ.ennrealToMeasure s = 0 → v s = 0
⊢ v ≪ᵥ μ

pp: case mpr
α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v : VectorMeasure α M
μ : VectorMeasure α ℝ≥0∞
h : v ≪ᵥ μ
⊢ ∀ ⦃s : Set α⦄, μ.ennrealToMeasure s = 0 → v s = 0

Tactic:
·

Goal state after:
case mp
α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v : VectorMeasure α M
μ : VectorMeasure α ℝ≥0∞
h : ∀ ⦃s : Set α⦄, μ.ennrealToMeasure s = 0 → v s = 0
⊢ v ≪ᵥ μ
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case mpr
α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v : VectorMeasure α M
μ : VectorMeasure α ℝ≥0∞
h : v ≪ᵥ μ
⊢ ∀ ⦃s : Set α⦄, μ.ennrealToMeasure s = 0 → v s = 0

Tactic:
·

Goal state after:
case mpr
α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v : VectorMeasure α M
μ : VectorMeasure α ℝ≥0∞
h : v ≪ᵥ μ
⊢ ∀ ⦃s : Set α⦄, μ.ennrealToMeasure s = 0 → v s = 0
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn

Goal state before:
pp: case refine_1
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
A : Set α := i \ ⋃ l, s.restrictNonposSeq i l
hA : A = i \ ⋃ l, s.restrictNonposSeq i l
bdd : ℕ → ℕ := fun n ↦ s.findExistsOneDivLT (i \ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k)
hn' : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l ≤ n), s.restrictNonposSeq i l] 0
h₁ : s i = s A + ∑' (l : ℕ), s (s.restrictNonposSeq i l)
h₂ : s A ≤ s i
this : Summable fun l ↦ s (s.restrictNonposSeq i l)
n : ℕ
⊢ 0 ≤ 1 / (↑(bdd n) + 1)

pp: case refine_2
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
A : Set α := i \ ⋃ l, s.restrictNonposSeq i l
hA : A = i \ ⋃ l, s.restrictNonposSeq i l
bdd : ℕ → ℕ := fun n ↦ s.findExistsOneDivLT (i \ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k)
hn' : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l ≤ n), s.restrictNonposSeq i l] 0
h₁ : s i = s A + ∑' (l : ℕ), s (s.restrictNonposSeq i l)
h₂ : s A ≤ s i
this : Summable fun l ↦ s (s.restrictNonposSeq i l)
n : ℕ
⊢ 1 / (↑(bdd n) + 1) ≤ ((fun l ↦ s (s.restrictNonposSeq i l)) ∘ Nat.succ) n

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
A : Set α := i \ ⋃ l, s.restrictNonposSeq i l
hA : A = i \ ⋃ l, s.restrictNonposSeq i l
bdd : ℕ → ℕ := fun n ↦ s.findExistsOneDivLT (i \ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k)
hn' : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l ≤ n), s.restrictNonposSeq i l] 0
h₁ : s i = s A + ∑' (l : ℕ), s (s.restrictNonposSeq i l)
h₂ : s A ≤ s i
this : Summable fun l ↦ s (s.restrictNonposSeq i l)
n : ℕ
⊢ 0 ≤ 1 / (↑(bdd n) + 1)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v : VectorMeasure α M
μ : VectorMeasure α ℝ≥0∞
h : v ≪ᵥ μ
s : Set α
hs : μ.ennrealToMeasure s = 0
hmeas : MeasurableSet s
⊢ v s = 0

pp: case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v : VectorMeasure α M
μ : VectorMeasure α ℝ≥0∞
h : v ≪ᵥ μ
s : Set α
hs : μ.ennrealToMeasure s = 0
hmeas : ¬MeasurableSet s
⊢ v s = 0

Tactic:
·

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v : VectorMeasure α M
μ : VectorMeasure α ℝ≥0∞
h : v ≪ᵥ μ
s : Set α
hs : μ.ennrealToMeasure s = 0
hmeas : MeasurableSet s
⊢ v s = 0
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v : VectorMeasure α M
μ : VectorMeasure α ℝ≥0∞
h : v ≪ᵥ μ
s : Set α
hs : μ.ennrealToMeasure s = 0
hmeas : ¬MeasurableSet s
⊢ v s = 0

Tactic:
·

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v : VectorMeasure α M
μ : VectorMeasure α ℝ≥0∞
h : v ≪ᵥ μ
s : Set α
hs : μ.ennrealToMeasure s = 0
hmeas : ¬MeasurableSet s
⊢ v s = 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✝ : 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 (f i)) p μ ≤ ENNReal.ofReal ε

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 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 (f i)) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn

Goal state before:
pp: case refine_2
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
A : Set α := i \ ⋃ l, s.restrictNonposSeq i l
hA : A = i \ ⋃ l, s.restrictNonposSeq i l
bdd : ℕ → ℕ := fun n ↦ s.findExistsOneDivLT (i \ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k)
hn' : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l ≤ n), s.restrictNonposSeq i l] 0
h₁ : s i = s A + ∑' (l : ℕ), s (s.restrictNonposSeq i l)
h₂ : s A ≤ s i
this : Summable fun l ↦ s (s.restrictNonposSeq i l)
n : ℕ
⊢ 1 / (↑(bdd n) + 1) ≤ ((fun l ↦ s (s.restrictNonposSeq i l)) ∘ Nat.succ) n

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi : s i < 0
hi₁ : MeasurableSet i
h : ¬s ≤[i] 0
hn : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0
A : Set α := i \ ⋃ l, s.restrictNonposSeq i l
hA : A = i \ ⋃ l, s.restrictNonposSeq i l
bdd : ℕ → ℕ := fun n ↦ s.findExistsOneDivLT (i \ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k)
hn' : ∀ (n : ℕ), ¬s ≤[i \ ⋃ l, ⋃ (_ : l ≤ n), s.restrictNonposSeq i l] 0
h₁ : s i = s A + ∑' (l : ℕ), s (s.restrictNonposSeq i l)
h₂ : s A ≤ s i
this : Summable fun l ↦ s (s.restrictNonposSeq i l)
n : ℕ
⊢ 1 / (↑(bdd n) + 1) ≤ ((fun l ↦ s (s.restrictNonposSeq i l)) ∘ Nat.succ) n
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
⊢ Disjoint (i ∩ S) (i ∩ Sᶜ)

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 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
⊢ Disjoint (i ∩ S) (i ∩ Sᶜ)