mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Covering.LiminfLimsup

Goal state before:
pp: case pos
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[>] 0) →
      blimsup (fun i ↦ cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i ↦ cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i ↦ if 0 < r i then r i else 1 / (↑i + 1)
i : ℕ
hi : 0 < r i
⊢ r' i ∈ Ioi 0

pp: case neg
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[>] 0) →
      blimsup (fun i ↦ cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i ↦ cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i ↦ if 0 < r i then r i else 1 / (↑i + 1)
i : ℕ
hi : ¬0 < r i
⊢ r' i ∈ Ioi 0

Tactic:
·

Goal state after:
case pos
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[>] 0) →
      blimsup (fun i ↦ cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i ↦ cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i ↦ if 0 < r i then r i else 1 / (↑i + 1)
i : ℕ
hi : 0 < r i
⊢ r' i ∈ Ioi 0
Module: Mathlib.MeasureTheory.Covering.LiminfLimsup

Goal state before:
pp: case neg
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[>] 0) →
      blimsup (fun i ↦ cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i ↦ cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i ↦ if 0 < r i then r i else 1 / (↑i + 1)
i : ℕ
hi : ¬0 < r i
⊢ r' i ∈ Ioi 0

Tactic:
·

Goal state after:
case neg
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[>] 0) →
      blimsup (fun i ↦ cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i ↦ cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i ↦ if 0 < r i then r i else 1 / (↑i + 1)
i : ℕ
hi : ¬0 < r i
⊢ r' i ∈ Ioi 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
f : α → β
hf : MemLp f ∞ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ∞
x : α
hx : x ∈ {x | (eLpNormEssSup f μ + 1).toReal ≤ ‖f x‖}
⊢ eLpNormEssSup f μ < eLpNormEssSup f μ + 1

pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hf : MemLp f ∞ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ∞
x : α
hx : x ∈ {x | (eLpNormEssSup f μ + 1).toReal ≤ ‖f x‖}
⊢ eLpNormEssSup f μ + 1 ≠ ∞

Tactic:
·

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hf : MemLp f ∞ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ∞
x : α
hx : x ∈ {x | (eLpNormEssSup f μ + 1).toReal ≤ ‖f x‖}
⊢ eLpNormEssSup f μ < eLpNormEssSup f μ + 1
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case hm
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
ι : Type u_4
s : Set ι
f : ι → Set α
hs : s.Countable
hd : s.Pairwise (Disjoint on f)
h : ∀ b ∈ s, MeasurableSet (f b)
this : Encodable ↑s
⊢ ∀ (i : ↑s), MeasurableSet (f ↑i)

pp: case hd
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
ι : Type u_4
s : Set ι
f : ι → Set α
hs : s.Countable
hd : s.Pairwise (Disjoint on f)
h : ∀ b ∈ s, MeasurableSet (f b)
this : Encodable ↑s
⊢ Pairwise (Disjoint on fun i ↦ f ↑i)

Tactic:
·

Goal state after:
case hm
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
ι : Type u_4
s : Set ι
f : ι → Set α
hs : s.Countable
hd : s.Pairwise (Disjoint on f)
h : ∀ b ∈ s, MeasurableSet (f b)
this : Encodable ↑s
⊢ ∀ (i : ↑s), MeasurableSet (f ↑i)
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
f : α → β
hf : MemLp f ∞ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ∞
x : α
hx : x ∈ {x | (eLpNormEssSup f μ + 1).toReal ≤ ‖f x‖}
⊢ eLpNormEssSup f μ + 1 ≠ ∞

Tactic:
·

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hf : MemLp f ∞ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ∞
x : α
hx : x ∈ {x | (eLpNormEssSup f μ + 1).toReal ≤ ‖f x‖}
⊢ eLpNormEssSup f μ + 1 ≠ ∞
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case hd
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
ι : Type u_4
s : Set ι
f : ι → Set α
hs : s.Countable
hd : s.Pairwise (Disjoint on f)
h : ∀ b ∈ s, MeasurableSet (f b)
this : Encodable ↑s
⊢ Pairwise (Disjoint on fun i ↦ f ↑i)

Tactic:
·

Goal state after:
case hd
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
ι : Type u_4
s : Set ι
f : ι → Set α
hs : s.Countable
hd : s.Pairwise (Disjoint on f)
h : ∀ b ∈ s, MeasurableSet (f b)
this : Encodable ↑s
⊢ Pairwise (Disjoint on fun i ↦ f ↑i)
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∞
f : α → β
hf : MemLp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : p = 0
⊢ ∃ M, eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) 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∞
f : α → β
hf : MemLp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
⊢ ∃ M, eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) 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∞
f : α → β
hf : MemLp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : p = 0
⊢ ∃ M, eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ ENNReal.ofReal ε
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∞
f : α → β
hf : MemLp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : p = ∞
⊢ ∃ M, eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) 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∞
f : α → β
hf : MemLp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ∞
⊢ ∃ M, eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) 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∞
f : α → β
hf : MemLp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : p = ∞
⊢ ∃ M, eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ ENNReal.ofReal ε
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h : A ⊆ B
⊢ v (B \ A) = v A + v (B \ A) - v A

Tactic:
add_sub_cancel_left

Goal state after:
α : Type u_1
m : MeasurableSpace α
M : Type u_4
inst✝² : AddCommGroup M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h : A ⊆ B
⊢ v (B \ A) = v (B \ A)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case h
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ Disjoint (A \ B) (A ∩ B)

pp: case hA
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ MeasurableSet (A \ B)

pp: case hB
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ MeasurableSet (A ∩ B)

Tactic:
·

Goal state after:
case h
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ Disjoint (A \ B) (A ∩ B)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case hA
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ MeasurableSet (A \ B)

pp: case hB
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ MeasurableSet (A ∩ B)

Tactic:
·

Goal state after:
case hA
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ MeasurableSet (A \ B)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case hB
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ MeasurableSet (A ∩ B)

Tactic:
·

Goal state after:
case hB
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ MeasurableSet (A ∩ B)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case h
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ Disjoint (A ∩ B) (B \ A)

pp: case hA
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ MeasurableSet (A ∩ B)

pp: case hB
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ MeasurableSet (B \ A)

Tactic:
·

Goal state after:
case h
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ Disjoint (A ∩ B) (B \ A)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case hA
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ MeasurableSet (A ∩ B)

pp: case hB
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ MeasurableSet (B \ A)

Tactic:
·

Goal state after:
case hA
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ MeasurableSet (A ∩ B)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case hB
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ MeasurableSet (B \ A)

Tactic:
·

Goal state after:
case hB
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ MeasurableSet (B \ A)
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∞
f : α → β
hf : MemLp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ∞
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ‖{x | M ≤ ↑‖‖f x‖ ^ p.toReal‖₊}.indicator (fun x ↦ ‖f x‖ ^ p.toReal) x‖ₑ ∂μ ≤
    ENNReal.ofReal (ε ^ p.toReal)
⊢ (∫⁻ (x : α), ‖{x | M ^ (1 / p.toReal) ≤ ↑‖f x‖₊}.indicator f x‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toReal) ≤
    ENNReal.ofReal ε ^ (p.toReal * (1 / p.toReal))

Tactic:
ENNReal.rpow_mul

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∞
f : α → β
hf : MemLp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ∞
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ‖{x | M ≤ ↑‖‖f x‖ ^ p.toReal‖₊}.indicator (fun x ↦ ‖f x‖ ^ p.toReal) x‖ₑ ∂μ ≤
    ENNReal.ofReal (ε ^ p.toReal)
⊢ (∫⁻ (x : α), ‖{x | M ^ (1 / p.toReal) ≤ ↑‖f x‖₊}.indicator f x‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toReal) ≤
    (ENNReal.ofReal ε ^ p.toReal) ^ (1 / p.toReal)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut

Goal state before:
pp: case neg.refine_8
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst✝¹ : CompleteSpace E
hm : m ≤ mΩ
inst✝ : IsFiniteMeasure μ
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (ω : Ω) ∂μ, ‖f ω‖ ≤ c
fs : ℕ → SimpleFunc Ω F := hf.approxBounded c
hfs_tendsto : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun x ↦ (fs x) ω) atTop (𝓝 (f ω))
hμ : ¬μ = 0
this✝ : (ae μ).NeBot
hc : 0 ≤ c
hfs_bound : ∀ (n : ℕ) (ω : Ω), ‖(fs n) ω‖ ≤ c
this : μ[fun ω ↦ (B (f ω)) (μ[g | m] ω) | m] = fun ω ↦ (B (f ω)) (μ[g | m] ω)
⊢ ∀ (n : ℕ), ∀ᵐ (x : Ω) ∂μ, ‖(B ((fs n) x)) (μ[g | m] x)‖ ≤ ‖B‖ * c * ‖μ[g | m] x‖

pp: case neg.refine_9
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst✝¹ : CompleteSpace E
hm : m ≤ mΩ
inst✝ : IsFiniteMeasure μ
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (ω : Ω) ∂μ, ‖f ω‖ ≤ c
fs : ℕ → SimpleFunc Ω F := hf.approxBounded c
hfs_tendsto : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun x ↦ (fs x) ω) atTop (𝓝 (f ω))
hμ : ¬μ = 0
this✝ : (ae μ).NeBot
hc : 0 ≤ c
hfs_bound : ∀ (n : ℕ) (ω : Ω), ‖(fs n) ω‖ ≤ c
this : μ[fun ω ↦ (B (f ω)) (μ[g | m] ω) | m] = fun ω ↦ (B (f ω)) (μ[g | m] ω)
⊢ ∀ (n : ℕ), μ[fun ω ↦ (B ((fs n) ω)) (g ω) | m] =ᵐ[μ] μ[fun ω ↦ (B ((fs n) ω)) (μ[g | m] ω) | m]

Tactic:
·

Goal state after:
case neg.refine_8
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst✝¹ : CompleteSpace E
hm : m ≤ mΩ
inst✝ : IsFiniteMeasure μ
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (ω : Ω) ∂μ, ‖f ω‖ ≤ c
fs : ℕ → SimpleFunc Ω F := hf.approxBounded c
hfs_tendsto : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun x ↦ (fs x) ω) atTop (𝓝 (f ω))
hμ : ¬μ = 0
this✝ : (ae μ).NeBot
hc : 0 ≤ c
hfs_bound : ∀ (n : ℕ) (ω : Ω), ‖(fs n) ω‖ ≤ c
this : μ[fun ω ↦ (B (f ω)) (μ[g | m] ω) | m] = fun ω ↦ (B (f ω)) (μ[g | m] ω)
⊢ ∀ (n : ℕ), ∀ᵐ (x : Ω) ∂μ, ‖(B ((fs n) x)) (μ[g | m] x)‖ ≤ ‖B‖ * c * ‖μ[g | m] x‖
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
s : ℕ → Set α
hm : Monotone s
hs : ∀ (i : ℕ), MeasurableSet (s i)
t : ℕ → Set α := disjointed s
ht : ∀ (n : ℕ), MeasurableSet (t n)
this : HasSum (fun n ↦ v (t n)) (v (⋃ n, s n))
n : ℕ
⊢ v (s n) = v (⋃ b ∈ Finset.range (n + 1), t b)

pp: case hd
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
s : ℕ → Set α
hm : Monotone s
hs : ∀ (i : ℕ), MeasurableSet (s i)
t : ℕ → Set α := disjointed s
ht : ∀ (n : ℕ), MeasurableSet (t n)
this : HasSum (fun n ↦ v (t n)) (v (⋃ n, s n))
n : ℕ
⊢ (↑(Finset.range (n + 1))).PairwiseDisjoint t

pp: case hm
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
s : ℕ → Set α
hm : Monotone s
hs : ∀ (i : ℕ), MeasurableSet (s i)
t : ℕ → Set α := disjointed s
ht : ∀ (n : ℕ), MeasurableSet (t n)
this : HasSum (fun n ↦ v (t n)) (v (⋃ n, s n))
n : ℕ
⊢ ∀ b ∈ Finset.range (n + 1), MeasurableSet (t b)

Tactic:
·

Goal state after:
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
s : ℕ → Set α
hm : Monotone s
hs : ∀ (i : ℕ), MeasurableSet (s i)
t : ℕ → Set α := disjointed s
ht : ∀ (n : ℕ), MeasurableSet (t n)
this : HasSum (fun n ↦ v (t n)) (v (⋃ n, s n))
n : ℕ
⊢ v (s n) = v (⋃ b ∈ Finset.range (n + 1), t b)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut

Goal state before:
pp: case neg.refine_9
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst✝¹ : CompleteSpace E
hm : m ≤ mΩ
inst✝ : IsFiniteMeasure μ
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (ω : Ω) ∂μ, ‖f ω‖ ≤ c
fs : ℕ → SimpleFunc Ω F := hf.approxBounded c
hfs_tendsto : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun x ↦ (fs x) ω) atTop (𝓝 (f ω))
hμ : ¬μ = 0
this✝ : (ae μ).NeBot
hc : 0 ≤ c
hfs_bound : ∀ (n : ℕ) (ω : Ω), ‖(fs n) ω‖ ≤ c
this : μ[fun ω ↦ (B (f ω)) (μ[g | m] ω) | m] = fun ω ↦ (B (f ω)) (μ[g | m] ω)
⊢ ∀ (n : ℕ), μ[fun ω ↦ (B ((fs n) ω)) (g ω) | m] =ᵐ[μ] μ[fun ω ↦ (B ((fs n) ω)) (μ[g | m] ω) | m]

Tactic:
·

Goal state after:
case neg.refine_9
Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst✝¹ : CompleteSpace E
hm : m ≤ mΩ
inst✝ : IsFiniteMeasure μ
f : Ω → F
g : Ω → E
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (ω : Ω) ∂μ, ‖f ω‖ ≤ c
fs : ℕ → SimpleFunc Ω F := hf.approxBounded c
hfs_tendsto : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun x ↦ (fs x) ω) atTop (𝓝 (f ω))
hμ : ¬μ = 0
this✝ : (ae μ).NeBot
hc : 0 ≤ c
hfs_bound : ∀ (n : ℕ) (ω : Ω), ‖(fs n) ω‖ ≤ c
this : μ[fun ω ↦ (B (f ω)) (μ[g | m] ω) | m] = fun ω ↦ (B (f ω)) (μ[g | m] ω)
⊢ ∀ (n : ℕ), μ[fun ω ↦ (B ((fs n) ω)) (g ω) | m] =ᵐ[μ] μ[fun ω ↦ (B ((fs n) ω)) (μ[g | m] ω) | m]
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case hd
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
s : ℕ → Set α
hm : Monotone s
hs : ∀ (i : ℕ), MeasurableSet (s i)
t : ℕ → Set α := disjointed s
ht : ∀ (n : ℕ), MeasurableSet (t n)
this : HasSum (fun n ↦ v (t n)) (v (⋃ n, s n))
n : ℕ
⊢ (↑(Finset.range (n + 1))).PairwiseDisjoint t

pp: case hm
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
s : ℕ → Set α
hm : Monotone s
hs : ∀ (i : ℕ), MeasurableSet (s i)
t : ℕ → Set α := disjointed s
ht : ∀ (n : ℕ), MeasurableSet (t n)
this : HasSum (fun n ↦ v (t n)) (v (⋃ n, s n))
n : ℕ
⊢ ∀ b ∈ Finset.range (n + 1), MeasurableSet (t b)

Tactic:
·

Goal state after:
case hd
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
s : ℕ → Set α
hm : Monotone s
hs : ∀ (i : ℕ), MeasurableSet (s i)
t : ℕ → Set α := disjointed s
ht : ∀ (n : ℕ), MeasurableSet (t n)
this : HasSum (fun n ↦ v (t n)) (v (⋃ n, s n))
n : ℕ
⊢ (↑(Finset.range (n + 1))).PairwiseDisjoint t
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case hm
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
s : ℕ → Set α
hm : Monotone s
hs : ∀ (i : ℕ), MeasurableSet (s i)
t : ℕ → Set α := disjointed s
ht : ∀ (n : ℕ), MeasurableSet (t n)
this : HasSum (fun n ↦ v (t n)) (v (⋃ n, s n))
n : ℕ
⊢ ∀ b ∈ Finset.range (n + 1), MeasurableSet (t b)

Tactic:
·

Goal state after:
case hm
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
s : ℕ → Set α
hm : Monotone s
hs : ∀ (i : ℕ), MeasurableSet (s i)
t : ℕ → Set α := disjointed s
ht : ∀ (n : ℕ), MeasurableSet (t n)
this : HasSum (fun n ↦ v (t n)) (v (⋃ n, s n))
n : ℕ
⊢ ∀ b ∈ Finset.range (n + 1), MeasurableSet (t b)