mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: case inl.inr
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
t u : Set Ω
ht : t.Finite
hs : ∅.Finite
hst : Disjoint ∅ t
ht' : t.Nonempty
⊢ (uniformOn ∅) u * (uniformOn (∅ ∪ t)) ∅ + (uniformOn t) u * (uniformOn (∅ ∪ t)) t = (uniformOn (∅ ∪ t)) u

pp: case inr.inl
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s u : Set Ω
hs : s.Finite
hs' : s.Nonempty
ht : ∅.Finite
hst : Disjoint s ∅
⊢ (uniformOn s) u * (uniformOn (s ∪ ∅)) s + (uniformOn ∅) u * (uniformOn (s ∪ ∅)) ∅ = (uniformOn (s ∪ ∅)) u

pp: case inr.inr
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ (uniformOn s) u * (uniformOn (s ∪ t)) s + (uniformOn t) u * (uniformOn (s ∪ t)) t = (uniformOn (s ∪ t)) u

Tactic:
·

Goal state after:
case inl.inr
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
t u : Set Ω
ht : t.Finite
hs : ∅.Finite
hst : Disjoint ∅ t
ht' : t.Nonempty
⊢ (uniformOn ∅) u * (uniformOn (∅ ∪ t)) ∅ + (uniformOn t) u * (uniformOn (∅ ∪ t)) t = (uniformOn (∅ ∪ t)) u
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: case inr.inl
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s u : Set Ω
hs : s.Finite
hs' : s.Nonempty
ht : ∅.Finite
hst : Disjoint s ∅
⊢ (uniformOn s) u * (uniformOn (s ∪ ∅)) s + (uniformOn ∅) u * (uniformOn (s ∪ ∅)) ∅ = (uniformOn (s ∪ ∅)) u

pp: case inr.inr
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ (uniformOn s) u * (uniformOn (s ∪ t)) s + (uniformOn t) u * (uniformOn (s ∪ t)) t = (uniformOn (s ∪ t)) u

Tactic:
·

Goal state after:
case inr.inl
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s u : Set Ω
hs : s.Finite
hs' : s.Nonempty
ht : ∅.Finite
hst : Disjoint s ∅
⊢ (uniformOn s) u * (uniformOn (s ∪ ∅)) s + (uniformOn ∅) u * (uniformOn (s ∪ ∅)) ∅ = (uniformOn (s ∪ ∅)) u
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: case inr.inr
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ (uniformOn s) u * (uniformOn (s ∪ t)) s + (uniformOn t) u * (uniformOn (s ∪ t)) t = (uniformOn (s ∪ t)) u

Tactic:
uniformOn,

Goal state after:
case inr.inr
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count[u | s] * (uniformOn (s ∪ t)) s + (uniformOn t) u * (uniformOn (s ∪ t)) t = (uniformOn (s ∪ t)) u
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: case inr.inr
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count[u | s] * (uniformOn (s ∪ t)) s + (uniformOn t) u * (uniformOn (s ∪ t)) t = (uniformOn (s ∪ t)) u

Tactic:
uniformOn,

Goal state after:
case inr.inr
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count[u | s] * Measure.count[s | s ∪ t] + (uniformOn t) u * Measure.count[t | s ∪ t] =
    Measure.count[u | s ∪ t]
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: case inr.inr
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count[u | s] * Measure.count[s | s ∪ t] + (uniformOn t) u * Measure.count[t | s ∪ t] =
    Measure.count[u | s ∪ t]

Tactic:
uniformOn,

Goal state after:
case inr.inr
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count[u | s] * Measure.count[s | s ∪ t] + Measure.count[u | t] * Measure.count[t | s ∪ t] =
    Measure.count[u | s ∪ t]
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case inl
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
s : (i : ι) → Set (α i)
h✝ : IsEmpty ι
⊢ piPremeasure m (univ.pi s) = ∏ i, (m i) (s i)

pp: case inr
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
s : (i : ι) → Set (α i)
h✝ : Nonempty ι
⊢ piPremeasure m (univ.pi s) = ∏ i, (m i) (s i)

Tactic:
·

Goal state after:
case inl
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
s : (i : ι) → Set (α i)
h✝ : IsEmpty ι
⊢ piPremeasure m (univ.pi s) = ∏ i, (m i) (s i)
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case inr.inl
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
s : (i : ι) → Set (α i)
h✝ : Nonempty ι
h : univ.pi s = ∅
⊢ piPremeasure m (univ.pi s) = ∏ i, (m i) (s i)

pp: case inr.inr
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
s : (i : ι) → Set (α i)
h✝ : Nonempty ι
h : (univ.pi s).Nonempty
⊢ piPremeasure m (univ.pi s) = ∏ i, (m i) (s i)

Tactic:
·

Goal state after:
case inr.inl
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
s : (i : ι) → Set (α i)
h✝ : Nonempty ι
h : univ.pi s = ∅
⊢ piPremeasure m (univ.pi s) = ∏ i, (m i) (s i)
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case inr.inr
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
s : (i : ι) → Set (α i)
h✝ : Nonempty ι
h : (univ.pi s).Nonempty
⊢ piPremeasure m (univ.pi s) = ∏ i, (m i) (s i)

Tactic:
·

Goal state after:
case inr.inr
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
s : (i : ι) → Set (α i)
h✝ : Nonempty ι
h : (univ.pi s).Nonempty
⊢ piPremeasure m (univ.pi s) = ∏ i, (m i) (s i)
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case inl
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
s : (i : ι) → Set (α i)
h : univ.pi s = ∅
⊢ (OuterMeasure.pi m) (univ.pi s) ≤ ∏ i, (m i) (s i)

pp: case inr
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
s : (i : ι) → Set (α i)
h : (univ.pi s).Nonempty
⊢ (OuterMeasure.pi m) (univ.pi s) ≤ ∏ i, (m i) (s i)

Tactic:
·

Goal state after:
case inl
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
s : (i : ι) → Set (α i)
h : univ.pi s = ∅
⊢ (OuterMeasure.pi m) (univ.pi s) ≤ ∏ i, (m i) (s i)
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case mp
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
n : OuterMeasure ((i : ι) → α i)
⊢ (∀ (s : Set ((i : ι) → α i)), s.Nonempty → n s ≤ piPremeasure m s) →
    ∀ (s : (i : ι) → Set (α i)), (univ.pi s).Nonempty → n (univ.pi s) ≤ ∏ i, (m i) (s i)

pp: case mpr
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
n : OuterMeasure ((i : ι) → α i)
⊢ (∀ (s : (i : ι) → Set (α i)), (univ.pi s).Nonempty → n (univ.pi s) ≤ ∏ i, (m i) (s i)) →
    ∀ (s : Set ((i : ι) → α i)), s.Nonempty → n s ≤ piPremeasure m s

Tactic:
·

Goal state after:
case mp
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
n : OuterMeasure ((i : ι) → α i)
⊢ (∀ (s : Set ((i : ι) → α i)), s.Nonempty → n s ≤ piPremeasure m s) →
    ∀ (s : (i : ι) → Set (α i)), (univ.pi s).Nonempty → n (univ.pi s) ≤ ∏ i, (m i) (s i)
Module: Mathlib.MeasureTheory.Function.EssSup

Goal state before:
pp: case inl
α : Type u_1
m : MeasurableSpace α
f : α → ℝ≥0∞
h₁ : ∀ᵐ (a : α) ∂0, f a ≠ ∞
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (ae 0) fun i ↦ (f i).toReal
⊢ (essSup f 0).toReal = essSup (fun a ↦ (f a).toReal) 0

pp: case inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h₁ : ∀ᵐ (a : α) ∂μ, f a ≠ ∞
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (ae μ) fun i ↦ (f i).toReal
hμ : NeZero μ
⊢ (essSup f μ).toReal = essSup (fun a ↦ (f a).toReal) μ

Tactic:
·

Goal state after:
case inl
α : Type u_1
m : MeasurableSpace α
f : α → ℝ≥0∞
h₁ : ∀ᵐ (a : α) ∂0, f a ≠ ∞
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (ae 0) fun i ↦ (f i).toReal
⊢ (essSup f 0).toReal = essSup (fun a ↦ (f a).toReal) 0
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case mpr
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
n : OuterMeasure ((i : ι) → α i)
⊢ (∀ (s : (i : ι) → Set (α i)), (univ.pi s).Nonempty → n (univ.pi s) ≤ ∏ i, (m i) (s i)) →
    ∀ (s : Set ((i : ι) → α i)), s.Nonempty → n s ≤ piPremeasure m s

Tactic:
·

Goal state after:
case mpr
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
n : OuterMeasure ((i : ι) → α i)
⊢ (∀ (s : (i : ι) → Set (α i)), (univ.pi s).Nonempty → n (univ.pi s) ≤ ∏ i, (m i) (s i)) →
    ∀ (s : Set ((i : ι) → α i)), s.Nonempty → n s ≤ piPremeasure m s
Module: Mathlib.MeasureTheory.Function.EssSup

Goal state before:
pp: case inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h₁ : ∀ᵐ (a : α) ∂μ, f a ≠ ∞
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (ae μ) fun i ↦ (f i).toReal
hμ : NeZero μ
⊢ (essSup f μ).toReal = essSup (fun a ↦ (f a).toReal) μ

Tactic:
·

Goal state after:
case inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
h₁ : ∀ᵐ (a : α) ∂μ, f a ≠ ∞
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (ae μ) fun i ↦ (f i).toReal
hμ : NeZero μ
⊢ (essSup f μ).toReal = essSup (fun a ↦ (f a).toReal) μ
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case refine_1
ι : Type u_1
α : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
i : ι
s : Set (α i)
hs : MeasurableSet s
t : Set ((a : ι) → α a)
⊢ (μ i).toOuterMeasure (eval i '' (t ∩ (fun b ↦ b i) ⁻¹' s)) +
      (μ i).toOuterMeasure (eval i '' (t \ (fun b ↦ b i) ⁻¹' s)) ≤
    (μ i).toOuterMeasure (eval i '' t)

pp: case refine_2
ι : Type u_1
α : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
i : ι
s : Set (α i)
hs : MeasurableSet s
t : Set ((a : ι) → α a)
⊢ ∀ j ∈ Finset.univ,
    j ≠ i → (μ j).toOuterMeasure (eval j '' (t ∩ (fun b ↦ b i) ⁻¹' s)) ≤ (μ j).toOuterMeasure (eval j '' t)

pp: case refine_3
ι : Type u_1
α : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
i : ι
s : Set (α i)
hs : MeasurableSet s
t : Set ((a : ι) → α a)
⊢ ∀ j ∈ Finset.univ,
    j ≠ i → (μ j).toOuterMeasure (eval j '' (t \ (fun b ↦ b i) ⁻¹' s)) ≤ (μ j).toOuterMeasure (eval j '' t)

Tactic:
·

Goal state after:
case refine_1
ι : Type u_1
α : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
i : ι
s : Set (α i)
hs : MeasurableSet s
t : Set ((a : ι) → α a)
⊢ (μ i).toOuterMeasure (eval i '' (t ∩ (fun b ↦ b i) ⁻¹' s)) +
      (μ i).toOuterMeasure (eval i '' (t \ (fun b ↦ b i) ⁻¹' s)) ≤
    (μ i).toOuterMeasure (eval i '' t)
Module: Mathlib.MeasureTheory.Function.SpecialFunctions.Basic

Goal state before:
pp: case refine_1
⊢ Measurable fun p ↦ (↑p.1, p.2).1 ^ (↑p.1, p.2).2

pp: case refine_2
⊢ Measurable fun x ↦ (∞, x).1 ^ (∞, x).2

Tactic:
·

Goal state after:
case refine_1
⊢ Measurable fun p ↦ (↑p.1, p.2).1 ^ (↑p.1, p.2).2
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case refine_2
ι : Type u_1
α : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
i : ι
s : Set (α i)
hs : MeasurableSet s
t : Set ((a : ι) → α a)
⊢ ∀ j ∈ Finset.univ,
    j ≠ i → (μ j).toOuterMeasure (eval j '' (t ∩ (fun b ↦ b i) ⁻¹' s)) ≤ (μ j).toOuterMeasure (eval j '' t)

pp: case refine_3
ι : Type u_1
α : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
i : ι
s : Set (α i)
hs : MeasurableSet s
t : Set ((a : ι) → α a)
⊢ ∀ j ∈ Finset.univ,
    j ≠ i → (μ j).toOuterMeasure (eval j '' (t \ (fun b ↦ b i) ⁻¹' s)) ≤ (μ j).toOuterMeasure (eval j '' t)

Tactic:
·

Goal state after:
case refine_2
ι : Type u_1
α : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
i : ι
s : Set (α i)
hs : MeasurableSet s
t : Set ((a : ι) → α a)
⊢ ∀ j ∈ Finset.univ,
    j ≠ i → (μ j).toOuterMeasure (eval j '' (t ∩ (fun b ↦ b i) ⁻¹' s)) ≤ (μ j).toOuterMeasure (eval j '' t)
Module: Mathlib.MeasureTheory.Function.SpecialFunctions.Basic

Goal state before:
pp: case refine_2
⊢ Measurable fun x ↦ (∞, x).1 ^ (∞, x).2

Tactic:
·

Goal state after:
case refine_2
⊢ Measurable fun x ↦ (∞, x).1 ^ (∞, x).2
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case refine_3
ι : Type u_1
α : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
i : ι
s : Set (α i)
hs : MeasurableSet s
t : Set ((a : ι) → α a)
⊢ ∀ j ∈ Finset.univ,
    j ≠ i → (μ j).toOuterMeasure (eval j '' (t \ (fun b ↦ b i) ⁻¹' s)) ≤ (μ j).toOuterMeasure (eval j '' t)

Tactic:
·

Goal state after:
case refine_3
ι : Type u_1
α : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
i : ι
s : Set (α i)
hs : MeasurableSet s
t : Set ((a : ι) → α a)
⊢ ∀ j ∈ Finset.univ,
    j ≠ i → (μ j).toOuterMeasure (eval j '' (t \ (fun b ↦ b i) ⁻¹' s)) ≤ (μ j).toOuterMeasure (eval j '' t)
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case refine_1
ι : Type u_1
α : ι → Type u_3
inst✝² : Fintype ι
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
s : (i : ι) → Set (α i)
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ (Measure.pi μ) (univ.pi s) ≤ ∏ i, (μ i) (s i)

pp: case refine_2
ι : Type u_1
α : ι → Type u_3
inst✝² : Fintype ι
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
s : (i : ι) → Set (α i)
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ ∏ i, (μ i) (s i) ≤ (Measure.pi μ) (univ.pi s)

Tactic:
·

Goal state after:
case refine_1
ι : Type u_1
α : ι → Type u_3
inst✝² : Fintype ι
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
s : (i : ι) → Set (α i)
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ (Measure.pi μ) (univ.pi s) ≤ ∏ i, (μ i) (s i)
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case refine_2
ι : Type u_1
α : ι → Type u_3
inst✝² : Fintype ι
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
s : (i : ι) → Set (α i)
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ ∏ i, (μ i) (s i) ≤ (Measure.pi μ) (univ.pi s)

Tactic:
·

Goal state after:
case refine_2
ι : Type u_1
α : ι → Type u_3
inst✝² : Fintype ι
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
s : (i : ι) → Set (α i)
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ ∏ i, (μ i) (s i) ≤ (Measure.pi μ) (univ.pi s)
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case refine_1
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝¹ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
C : (i : ι) → Set (Set (α i))
hμ : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)
this✝ : ∀ (i : ι), SigmaFinite (μ i)
this : Encodable ι
n : ℕ
e : ℕ → ι → ℕ := fun n ↦ (decode n).getD default
⊢ (univ.pi fun i ↦ (hμ i).set ((decode n).getD default i)) ∈ univ.pi '' univ.pi C

pp: case refine_2
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝¹ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
C : (i : ι) → Set (Set (α i))
hμ : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)
this✝ : ∀ (i : ι), SigmaFinite (μ i)
this : Encodable ι
n : ℕ
e : ℕ → ι → ℕ := fun n ↦ (decode n).getD default
⊢ (Measure.pi μ) (univ.pi fun i ↦ (hμ i).set ((decode n).getD default i)) < ∞

pp: case refine_3
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝¹ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
C : (i : ι) → Set (Set (α i))
hμ : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)
this✝ : ∀ (i : ι), SigmaFinite (μ i)
this : Encodable ι
e : ℕ → ι → ℕ := fun n ↦ (decode n).getD default
⊢ (⋃ i, univ.pi fun i_1 ↦ (hμ i_1).set ((decode i).getD default i_1)) = univ

Tactic:
·

Goal state after:
case refine_1
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝¹ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
C : (i : ι) → Set (Set (α i))
hμ : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)
this✝ : ∀ (i : ι), SigmaFinite (μ i)
this : Encodable ι
n : ℕ
e : ℕ → ι → ℕ := fun n ↦ (decode n).getD default
⊢ (univ.pi fun i ↦ (hμ i).set ((decode n).getD default i)) ∈ univ.pi '' univ.pi C
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case refine_2
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝¹ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
C : (i : ι) → Set (Set (α i))
hμ : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)
this✝ : ∀ (i : ι), SigmaFinite (μ i)
this : Encodable ι
n : ℕ
e : ℕ → ι → ℕ := fun n ↦ (decode n).getD default
⊢ (Measure.pi μ) (univ.pi fun i ↦ (hμ i).set ((decode n).getD default i)) < ∞

pp: case refine_3
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝¹ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
C : (i : ι) → Set (Set (α i))
hμ : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)
this✝ : ∀ (i : ι), SigmaFinite (μ i)
this : Encodable ι
e : ℕ → ι → ℕ := fun n ↦ (decode n).getD default
⊢ (⋃ i, univ.pi fun i_1 ↦ (hμ i_1).set ((decode i).getD default i_1)) = univ

Tactic:
·

Goal state after:
case refine_2
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝¹ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
C : (i : ι) → Set (Set (α i))
hμ : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)
this✝ : ∀ (i : ι), SigmaFinite (μ i)
this : Encodable ι
n : ℕ
e : ℕ → ι → ℕ := fun n ↦ (decode n).getD default
⊢ (Measure.pi μ) (univ.pi fun i ↦ (hμ i).set ((decode n).getD default i)) < ∞
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case refine_3
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝¹ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
C : (i : ι) → Set (Set (α i))
hμ : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)
this✝ : ∀ (i : ι), SigmaFinite (μ i)
this : Encodable ι
e : ℕ → ι → ℕ := fun n ↦ (decode n).getD default
⊢ (⋃ i, univ.pi fun i_1 ↦ (hμ i_1).set ((decode i).getD default i_1)) = univ

Tactic:
·

Goal state after:
case refine_3
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝¹ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
C : (i : ι) → Set (Set (α i))
hμ : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)
this✝ : ∀ (i : ι), SigmaFinite (μ i)
this : Encodable ι
e : ℕ → ι → ℕ := fun n ↦ (decode n).getD default
⊢ (⋃ i, univ.pi fun i_1 ↦ (hμ i_1).set ((decode i).getD default i_1)) = univ
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case refine_2
ι : Type u_1
α : ι → Type u_3
inst✝³ : Fintype ι
inst✝² : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝¹ : ∀ (i : ι), SigmaFinite (μ i)
inst✝ : DecidableEq ι
i : ι
s : Set (α i)
hs : MeasurableSet s
⊢ (μ i) (update (fun j ↦ univ) i s i) = (μ i) s

pp: case refine_1
ι : Type u_1
α : ι → Type u_3
inst✝³ : Fintype ι
inst✝² : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝¹ : ∀ (i : ι), SigmaFinite (μ i)
inst✝ : DecidableEq ι
i : ι
s : Set (α i)
hs : MeasurableSet s
⊢ ∏ x ∈ Finset.univ.erase i, (μ x) (update (fun j ↦ univ) i s x) = ∏ j ∈ Finset.univ.erase i, (μ j) univ

Tactic:
·

Goal state after:
case refine_2
ι : Type u_1
α : ι → Type u_3
inst✝³ : Fintype ι
inst✝² : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝¹ : ∀ (i : ι), SigmaFinite (μ i)
inst✝ : DecidableEq ι
i : ι
s : Set (α i)
hs : MeasurableSet s
⊢ (μ i) (update (fun j ↦ univ) i s i) = (μ i) s
Module: Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov

Goal state before:
pp: α : Type u_1
ε' : Type u_3
m0 : MeasurableSpace α
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
p : ℝ≥0∞
μ : Measure α
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
f : α → ε'
hf : AEStronglyMeasurable f μ
ε : ℝ≥0∞
this : 1 / p.toReal * p.toReal = 1
⊢ (ε * μ {x | ε ≤ ‖f x‖ₑ ^ p.toReal}) ^ (1 / p.toReal * p.toReal) ≤ eLpNorm f p μ ^ p.toReal

Tactic:
ENNReal.rpow_mul

Goal state after:
α : Type u_1
ε' : Type u_3
m0 : MeasurableSpace α
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
p : ℝ≥0∞
μ : Measure α
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
f : α → ε'
hf : AEStronglyMeasurable f μ
ε : ℝ≥0∞
this : 1 / p.toReal * p.toReal = 1
⊢ ((ε * μ {x | ε ≤ ‖f x‖ₑ ^ p.toReal}) ^ (1 / p.toReal)) ^ p.toReal ≤ eLpNorm f p μ ^ p.toReal
Module: Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov

Goal state before:
pp: case pos
α : Type u_1
ε' : Type u_3
m0 : MeasurableSpace α
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
p : ℝ≥0∞
μ : Measure α
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
f : α → ε'
hf : AEStronglyMeasurable f μ
ε : ℝ≥0∞
hε : ε ≠ 0
hmeas_top : ε = ∞ → μ {x | ‖f x‖ₑ = ∞} = 0
h : ε = ∞
⊢ μ {x | ε ≤ ‖f x‖ₑ} ≤ ε⁻¹ ^ p.toReal * eLpNorm f p μ ^ p.toReal

pp: case neg
α : Type u_1
ε' : Type u_3
m0 : MeasurableSpace α
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
p : ℝ≥0∞
μ : Measure α
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
f : α → ε'
hf : AEStronglyMeasurable f μ
ε : ℝ≥0∞
hε : ε ≠ 0
hmeas_top : ε = ∞ → μ {x | ‖f x‖ₑ = ∞} = 0
h : ¬ε = ∞
⊢ μ {x | ε ≤ ‖f x‖ₑ} ≤ ε⁻¹ ^ p.toReal * eLpNorm f p μ ^ p.toReal

Tactic:
·

Goal state after:
case pos
α : Type u_1
ε' : Type u_3
m0 : MeasurableSpace α
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
p : ℝ≥0∞
μ : Measure α
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
f : α → ε'
hf : AEStronglyMeasurable f μ
ε : ℝ≥0∞
hε : ε ≠ 0
hmeas_top : ε = ∞ → μ {x | ‖f x‖ₑ = ∞} = 0
h : ε = ∞
⊢ μ {x | ε ≤ ‖f x‖ₑ} ≤ ε⁻¹ ^ p.toReal * eLpNorm f p μ ^ p.toReal
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
X : ι → Type u_4
Y : ι → Type u_5
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
inst✝ : (i : ι) → MeasurableSpace (Y i)
f : (i : ι) → X i → Y i
hμ : ∀ (i : ι), SigmaFinite (map (f i) (μ i))
hf : ∀ (i : ι), AEMeasurable (f i) (μ i)
this : ∀ (i : ι), SigmaFinite (μ i)
s : (i : ι) → Set (Y i)
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ AEMeasurable (fun x i ↦ f i (x i)) (Measure.pi μ)

pp: ι : Type u_1
inst✝¹ : Fintype ι
X : ι → Type u_4
Y : ι → Type u_5
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
inst✝ : (i : ι) → MeasurableSpace (Y i)
f : (i : ι) → X i → Y i
hμ : ∀ (i : ι), SigmaFinite (map (f i) (μ i))
hf : ∀ (i : ι), AEMeasurable (f i) (μ i)
this : ∀ (i : ι), SigmaFinite (μ i)
s : (i : ι) → Set (Y i)
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ (Measure.pi μ) ((fun x i ↦ f i (x i)) ⁻¹' univ.pi s) = ∏ i, (map (f i) (μ i)) (s i)

Tactic:
·

Goal state after:
ι : Type u_1
inst✝¹ : Fintype ι
X : ι → Type u_4
Y : ι → Type u_5
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
inst✝ : (i : ι) → MeasurableSpace (Y i)
f : (i : ι) → X i → Y i
hμ : ∀ (i : ι), SigmaFinite (map (f i) (μ i))
hf : ∀ (i : ι), AEMeasurable (f i) (μ i)
this : ∀ (i : ι), SigmaFinite (μ i)
s : (i : ι) → Set (Y i)
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ AEMeasurable (fun x i ↦ f i (x i)) (Measure.pi μ)