mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: ι : Type u_1
inst✝² : LinearOrder ι
inst✝¹ : TopologicalSpace ι
inst✝ : OrderTopology ι
a : ↑{a | a ≠ ⊤}
⊢ Tendsto (fun x ↦ (↑x).untop ⋯) (𝓝 a) (𝓝 ((↑a).untop ⋯))

Tactic:
have : Nonempty ι := ⟨untop a.1 a.2⟩

Goal state after:
ι : Type u_1
inst✝² : LinearOrder ι
inst✝¹ : TopologicalSpace ι
inst✝ : OrderTopology ι
a : ↑{a | a ≠ ⊤}
this : Nonempty ι
⊢ Tendsto (fun x ↦ (↑x).untop ⋯) (𝓝 a) (𝓝 ((↑a).untop ⋯))
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) = ∑ i ∈ Finset.range (n + 1), v (t i)

Tactic:
← of_biUnion_finset

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)

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

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)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : MeasurableSpace β
x✝ : β
v✝ : M
s : Set β
x : β
v : M
f : ℕ → Set β
f_meas : ∀ (i : ℕ), MeasurableSet (f i)
f_disj : Pairwise (Disjoint on f)
hx : x ∈ ⋃ i, f i
⊢ MeasurableSet (⋃ i, f i)

Tactic:
apply MeasurableSet.iUnion f_meas

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case neg
α : Type u_1
mα : MeasurableSpace α
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v : VectorMeasure α M
i : Set α
hi : ¬MeasurableSet i
⊢ (v.restrict i) univ = v i

Tactic:
· simp [restrict_not_measurable, hi]

Goal state after:
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∞
f : α → β
hp_one : 1 ≤ p
hp_top : p ≠ ∞
hf : MemLp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
M : ℝ
hMpos : 0 < M
hM : eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ ENNReal.ofReal ε
δ : ℝ
hδpos : 0 < δ
hδ :
  ∀ (s : Set α),
    MeasurableSet s →
      ?m.82 s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator ({x | ‖f x‖ < M}.indicator f)) p ?m.82 ≤ ENNReal.ofReal ε
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal δ
⊢ eLpNorm (s.indicator ({x | M ≤ ↑‖f x‖₊}.indicator f + {x | ‖f x‖ < M}.indicator f)) p μ ≤ 2 * ENNReal.ofReal ε

Tactic:
rw [eLpNorm_indicator_eq_eLpNorm_restrict hs]
    refine le_trans (eLpNorm_add_le ?_ ?_ hp_one) ?_
    · exact StronglyMeasurable.aestronglyMeasurable
        (hmeas.indicator (measurableSet_le measurable_const hmeas.nnnorm.measurable.subtype_coe))
    · exact StronglyMeasurable.aestronglyMeasurable
        (hmeas.indicator (measurableSet_lt hmeas.nnnorm.measurable.subtype_coe measurable_const))
    · rw [two_mul]
      refine add_le_add (le_trans (eLpNorm_mono_measure _ Measure.restrict_le_self) hM) ?_
      rw [← eLpNorm_indicator_eq_eLpNorm_restrict hs]
      exact hδ s hs hμs

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommGroup M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : IsTopologicalAddGroup M
v w : VectorMeasure α M
i : Set α
hi : MeasurableSet i
h : v ≤[i] w
j : Set α
hj₁ : MeasurableSet j
⊢ v (j ∩ i) ≤ w (j ∩ i)

Tactic:
← restrict_apply _ hi hj₁,

Goal state after:
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommGroup M
inst✝² : PartialOrder M
inst✝¹ : IsOrderedAddMonoid M
inst✝ : IsTopologicalAddGroup M
v w : VectorMeasure α M
i : Set α
hi : MeasurableSet i
h : v ≤[i] w
j : Set α
hj₁ : MeasurableSet j
⊢ (v.restrict i) j ≤ w (j ∩ i)
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan

Goal state before:
pp: case refine_1
α : Type u_1
inst✝ : MeasurableSpace α
j : JordanDecomposition α
S : Set α
hS₁ : MeasurableSet S
hS₂ : j.posPart S = 0
hS₃ : j.negPart Sᶜ = 0
A : Set α
hA : MeasurableSet A
hA₁ : A ⊆ S
⊢ 0 ≤ j.negPart.real A

Tactic:
exact ENNReal.toReal_nonneg

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan

Goal state before:
pp: case refine_2
α : Type u_1
inst✝ : MeasurableSpace α
j : JordanDecomposition α
S : Set α
hS₁ : MeasurableSet S
hS₂ : j.posPart S = 0
hS₃ : j.negPart Sᶜ = 0
A : Set α
hA : MeasurableSet A
hA₁ : A ⊆ Sᶜ
⊢ 0 A ≤ j.posPart.real A

Tactic:
exact ENNReal.toReal_nonneg

Goal state after:
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

Tactic:
top_le_iff

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 mpr
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
μ : VectorMeasure α ℝ≥0∞
h : s.totalVariation ≪ μ.ennrealToMeasure
⊢ s ≪ᵥ μ

Tactic:
refine VectorMeasure.AbsolutelyContinuous.mk fun S hS₁ hS₂ => ?_
    rw [← VectorMeasure.ennrealToMeasure_apply hS₁] at hS₂
    exact null_of_totalVariation_zero s (h hS₂)

Goal state after:
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: 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
hε' : 0 < ε.toReal / 3
hdivp : 0 ≤ 1 / p.toReal
hpow : 0 < measureUnivNNReal μ ^ (1 / p.toReal)
δ₁ : ℝ
hδ₁ : 0 < δ₁
heLpNorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₁ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
heLpNorm₂ :
  ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ₂ → eLpNorm (s.indicator g) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
t : Set α
htm : MeasurableSet t
ht₁ : μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N n : ℕ
hN : ∀ x ∈ tᶜ, dist (g x) (f n x) < ε.toReal / (3 * ↑(measureUnivNNReal μ) ^ (1 / p.toReal))
hn : n ≥ N
hnf : eLpNorm (t.indicator (f n)) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
hng : eLpNorm (t.indicator g) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
⊢ eLpNorm (tᶜ.indicator (f n - g)) p μ ≤ ENNReal.ofReal (ε.toReal / 3)

Tactic:
have : 0 ≤ ε.toReal / (3 * measureUnivNNReal μ ^ (1 / p.toReal)) := by positivity

Goal state after:
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
hε' : 0 < ε.toReal / 3
hdivp : 0 ≤ 1 / p.toReal
hpow : 0 < measureUnivNNReal μ ^ (1 / p.toReal)
δ₁ : ℝ
hδ₁ : 0 < δ₁
heLpNorm₁ :
  ∀ (i : ℕ) (s : Set α),
    MeasurableSet s → μ s ≤ ENNReal.ofReal δ₁ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
δ₂ : ℝ
hδ₂ : 0 < δ₂
heLpNorm₂ :
  ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ₂ → eLpNorm (s.indicator g) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
t : Set α
htm : MeasurableSet t
ht₁ : μ t ≤ ENNReal.ofReal (min δ₁ δ₂)
N n : ℕ
hN : ∀ x ∈ tᶜ, dist (g x) (f n x) < ε.toReal / (3 * ↑(measureUnivNNReal μ) ^ (1 / p.toReal))
hn : n ≥ N
hnf : eLpNorm (t.indicator (f n)) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
hng : eLpNorm (t.indicator g) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
this : 0 ≤ ε.toReal / (3 * ↑(measureUnivNNReal μ) ^ (1 / p.toReal))
⊢ eLpNorm (tᶜ.indicator (f n - g)) p μ ≤ ENNReal.ofReal (ε.toReal / 3)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case e'_5
α : Type u_1
β : Type u_2
m✝ : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
m n✝ : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n✝
f : ℕ → Set α
hf₁ : ∀ (i : ℕ), MeasurableSet (f i)
hf₂ : Pairwise (Disjoint on f)
hf₁' : ∀ (k : ℕ), MeasurableSet (f k)
n : ℕ
⊢ (if MeasurableSet (f n) then v (f n) else 0) = v (f n)

Tactic:
if_pos (hf₁ n)

Goal state after:
case e'_5
α : Type u_1
β : Type u_2
m✝ : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
m n✝ : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n✝
f : ℕ → Set α
hf₁ : ∀ (i : ℕ), MeasurableSet (f i)
hf₂ : Pairwise (Disjoint on f)
hf₁' : ∀ (k : ℕ), MeasurableSet (f k)
n : ℕ
⊢ v (f n) = v (f n)
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: case e'_5
α : Type u_1
β : Type u_2
m✝ : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
m n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
f : ℕ → Set α
hf₁ : ∀ (i : ℕ), MeasurableSet (f i)
hf₂ : Pairwise (Disjoint on f)
hf₁' : ∀ (k : ℕ), MeasurableSet (f k)
⊢ (fun i ↦ if MeasurableSet (f i) then v (f i) else 0) = fun i ↦ v (f i)

pp: case e'_6
α : Type u_1
β : Type u_2
m✝ : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
m n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
f : ℕ → Set α
hf₁ : ∀ (i : ℕ), MeasurableSet (f i)
hf₂ : Pairwise (Disjoint on f)
hf₁' : ∀ (k : ℕ), MeasurableSet (f k)
⊢ (if MeasurableSet (⋃ i, f i) then v (⋃ i, f i) else 0) = v (⋃ i, f i)

Tactic:
· ext n
        rw [if_pos (hf₁ n)]

Goal state after:
case e'_6
α : Type u_1
β : Type u_2
m✝ : MeasurableSpace α
L : Type u_3
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid L
inst✝⁴ : TopologicalSpace L
inst✝³ : AddCommMonoid M
inst✝² : TopologicalSpace M
inst✝¹ : AddCommMonoid N
inst✝ : TopologicalSpace N
m n : MeasurableSpace α
v : VectorMeasure α M
hle : m ≤ n
f : ℕ → Set α
hf₁ : ∀ (i : ℕ), MeasurableSet (f i)
hf₂ : Pairwise (Disjoint on f)
hf₁' : ∀ (k : ℕ), MeasurableSet (f k)
⊢ (if MeasurableSet (⋃ i, f i) then v (⋃ i, f i) else 0) = v (⋃ i, f i)
Module: Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm

Goal state before:
pp: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f g : α → E
hf : MemLp f p μ
hp : 1 ≤ p
hg : ¬MemLp g p μ
⊢ lpNorm (f + g) p μ ≤ lpNorm f p μ + lpNorm g p μ

Tactic:
· rw [lpNorm_of_not_memLp fun hfg ↦ hg <| by simpa using hfg.sub hf, lpNorm_of_not_memLp hg]
    simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
g : α → ℝ
hg : MemLp g p μ
h : ∀ (x : α), ‖f x‖ ≤ g x
hf : AEStronglyMeasurable f μ
⊢ lpNorm f p μ ≤ lpNorm g p μ

pp: case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
g : α → ℝ
hg : MemLp g p μ
h : ∀ (x : α), ‖f x‖ ≤ g x
hf : ¬AEStronglyMeasurable f μ
⊢ lpNorm f p μ ≤ lpNorm g p μ

Tactic:
· rw [← toReal_eLpNorm hf, ← toReal_eLpNorm hg.aestronglyMeasurable]
    exact ENNReal.toNNReal_mono (hg.eLpNorm_ne_top) (eLpNorm_mono_real h)

Goal state after:
case neg
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
g : α → ℝ
hg : MemLp g p μ
h : ∀ (x : α), ‖f x‖ ≤ g x
hf : ¬AEStronglyMeasurable f μ
⊢ lpNorm f p μ ≤ lpNorm g p μ
Module: Mathlib.MeasureTheory.Measure.HasOuterApproxClosed

Goal state before:
pp: Ω : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
F : Set Ω
δ : ℝ
δ_pos : 0 < δ
⊢ Integrable (fun ω ↦ ↑((thickenedIndicator δ_pos F) ω)) μ

Tactic:
refine .of_bound (by fun_prop) 1 (ae_of_all _ fun x ↦ ?_)

Goal state after:
Ω : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
F : Set Ω
δ : ℝ
δ_pos : 0 < δ
x : Ω
⊢ ‖↑((thickenedIndicator δ_pos F) x)‖ ≤ 1
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ι → α → β
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hpzero : p ≠ 0
hμ : ¬μ univ = 0
ε : ℝ
hε : 0 < ε
C : ℝ≥0
hCpos : 0 < C
hC : ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal (ε / 2)
i : ι
s : Set α
hs : MeasurableSet s
hμs : μ s ≤ ENNReal.ofReal ((ε / (2 * ↑C)) ^ p.toReal)
hμs' : ¬μ s = 0
⊢ ENNReal.ofReal (ε / 2) + ↑C * ENNReal.ofReal (ε / (2 * ↑C)) ≤ ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2)

Tactic:
by
      gcongr
      rw [← ENNReal.ofReal_coe_nnreal, ← ENNReal.ofReal_mul (NNReal.coe_nonneg _), ← div_div,
        mul_div_cancel₀ _ (NNReal.coe_pos.2 hCpos).ne.symm]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.ProbabilityMeasure

Goal state before:
pp: Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
X : Type u_2
inst✝ : TopologicalSpace X
μs : X → ProbabilityMeasure Ω
⊢ Continuous[inst✝, instTopologicalSpace] μs ↔ ∀ (f : Ω →ᵇ ℝ), Continuous[inst✝, _] fun x ↦ ∫ (ω : Ω), f ω ∂↑(μs x)

Tactic:
by
  simp [continuous_iff_continuousAt, ContinuousAt, tendsto_iff_forall_integral_tendsto,
    forall_comm (α := X)]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.FiniteMeasure

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f₁ : Ω →ᵇ ℝ≥0
⊢ ∀ (y : Ω →ᵇ ℝ≥0), dist (μ.testAgainstNN f₁) (μ.testAgainstNN y) ≤ ↑μ.mass * dist f₁ y

Tactic:
f₂

Goal state after:
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f₁ f₂ : Ω →ᵇ ℝ≥0
⊢ dist (μ.testAgainstNN f₁) (μ.testAgainstNN f₂) ≤ ↑μ.mass * dist f₁ f₂
Module: Mathlib.MeasureTheory.Measure.ProbabilityMeasure

Goal state before:
pp: Ω : Type u_1
inst✝² : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μs_lim : Tendsto μs F (𝓝 μ)
nonzero : μ ≠ 0
f : Ω →ᵇ ℝ≥0
lim_mass : Tendsto (fun i ↦ (μs i).mass) F (𝓝 μ.mass)
aux : {0}ᶜ ∈ 𝓝 μ.mass
⊢ Tendsto (fun i ↦ (μs i).normalize.toFiniteMeasure.testAgainstNN f) F (𝓝 (μ.normalize.toFiniteMeasure.testAgainstNN f))

Tactic:
have eventually_nonzero : ∀ᶠ i in F, μs i ≠ 0 := by
    simp_rw [← mass_nonzero_iff]
    exact lim_mass aux

Goal state after:
Ω : Type u_1
inst✝² : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μs_lim : Tendsto μs F (𝓝 μ)
nonzero : μ ≠ 0
f : Ω →ᵇ ℝ≥0
lim_mass : Tendsto (fun i ↦ (μs i).mass) F (𝓝 μ.mass)
aux : {0}ᶜ ∈ 𝓝 μ.mass
eventually_nonzero : ∀ᶠ (i : γ) in F, μs i ≠ 0
⊢ Tendsto (fun i ↦ (μs i).normalize.toFiniteMeasure.testAgainstNN f) F (𝓝 (μ.normalize.toFiniteMeasure.testAgainstNN f))
Module: Mathlib.MeasureTheory.Measure.FiniteMeasure

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_3
F : Filter γ
μs : γ → FiniteMeasure Ω
mass_lim : Tendsto (fun i ↦ (μs i).mass) F (𝓝 0)
f : Ω →ᵇ ℝ≥0
⊢ Tendsto (fun i ↦ (μs i).testAgainstNN f) F (𝓝 0)

Tactic:
by
  apply tendsto_iff_dist_tendsto_zero.mpr
  have obs := fun i ↦ (μs i).testAgainstNN_lipschitz_estimate f 0
  simp_rw [testAgainstNN_zero, zero_add] at obs
  simp_rw [show ∀ i, dist ((μs i).testAgainstNN f) 0 = (μs i).testAgainstNN f by
      simp only [dist_nndist, NNReal.nndist_zero_eq_val', imp_true_iff]]
  apply squeeze_zero (fun i ↦ NNReal.coe_nonneg _) obs
  have lim_pair : Tendsto (fun i ↦ (⟨nndist f 0, (μs i).mass⟩ : ℝ × ℝ)) F (𝓝 ⟨nndist f 0, 0⟩) :=
    (Prod.tendsto_iff _ _).mpr ⟨tendsto_const_nhds, (NNReal.continuous_coe.tendsto 0).comp mass_lim⟩
  simpa using! tendsto_mul.comp lim_pair

Goal state after:
Module: Mathlib.MeasureTheory.Measure.ProbabilityMeasure

Goal state before:
pp: Ω : Type u_1
inst✝² : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
μs_lim : Tendsto (fun i ↦ (μs i).normalize) F (𝓝 μ.normalize)
mass_lim : Tendsto (fun i ↦ (μs i).mass) F (𝓝 μ.mass)
⊢ Tendsto μs F (𝓝 μ)

Tactic:
by
  rw [tendsto_iff_forall_testAgainstNN_tendsto]
  exact fun f ↦
    tendsto_testAgainstNN_of_tendsto_normalize_testAgainstNN_of_tendsto_mass μs_lim mass_lim f

Goal state after:
Module: Mathlib.MeasureTheory.Measure.ProbabilityMeasure

Goal state before:
pp: Ω : Type u_1
inst✝² : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
nonzero : μ ≠ 0
⊢ Tendsto (fun i ↦ (μs i).normalize) F (𝓝 μ.normalize) ∧ Tendsto (fun i ↦ (μs i).mass) F (𝓝 μ.mass) ↔ Tendsto μs F (𝓝 μ)

Tactic:
constructor
  · rintro ⟨normalized_lim, mass_lim⟩
    exact tendsto_of_tendsto_normalize_testAgainstNN_of_tendsto_mass normalized_lim mass_lim
  · intro μs_lim
    exact ⟨tendsto_normalize_of_tendsto μs_lim nonzero, μs_lim.mass⟩

Goal state after:
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ∞
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), eLpNorm (f i) p μ ≤ ↑M
ℐ : ℝ≥0 → ι
δ : ℝ≥0
hδpos : 0 < ↑δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal ↑δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hℐ : ∀ (C : ℝ≥0), ENNReal.ofReal ↑δ < μ {x | C ≤ ‖f (ℐ C) x‖₊}
this : ↑2 * ↑(max M 1) ≤ eLpNorm (f (ℐ (2 * max M 1 * δ⁻¹ ^ (1 / p.toReal)))) p μ
⊢ max (↑M) 1 < ↑2 * ↑(max M 1)

Tactic:
rw [← ENNReal.coe_one, ← ENNReal.coe_max, ← ENNReal.coe_mul, ENNReal.coe_lt_coe]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
hp : p ≠ 0
hp' : p ≠ ∞
hf : ∀ (i : ι), StronglyMeasurable (f i)
ε : ℝ
hε : 0 < ε
hfu : UnifIntegrable f p μ
M : ℝ≥0
hM : ∀ (i : ι), eLpNorm (f i) p μ ≤ ↑M
ℐ : ℝ≥0 → ι
δ : ℝ≥0
hδpos : 0 < ↑δ
hδ :
  ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal ↑δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε
hℐ : ∀ (C : ℝ≥0), ENNReal.ofReal ↑δ < μ {x | C ≤ ‖f (ℐ C) x‖₊}
this : ↑2 * ↑(max M 1) ≤ eLpNorm (f (ℐ (2 * max M 1 * δ⁻¹ ^ (1 / p.toReal)))) p μ
⊢ max M 1 < 2 * max M 1
Module: Mathlib.MeasureTheory.Measure.FiniteMeasure

Goal state before:
pp: Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
X : Type u_2
inst✝ : TopologicalSpace X
μs : X → FiniteMeasure Ω
⊢ Continuous[inst✝, instTopologicalSpace] μs ↔ ∀ (f : Ω →ᵇ ℝ), Continuous[inst✝, _] fun x ↦ ∫ (ω : Ω), f ω ∂↑(μs x)

Tactic:
by
  simp [continuous_iff_continuousAt, ContinuousAt, tendsto_iff_forall_integral_tendsto,
    forall_comm (α := X)]

Goal state after:
Module: Mathlib.Probability.Independence.Kernel.Indep

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
s : ι → Set (Set Ω)
S T : Set ι
h_indep : iIndepSets s κ μ
hST : Disjoint S T
t1 t2 : Set Ω
p1 : Finset ι
hp1 : ↑p1 ⊆ S
f1 : ι → Set Ω
ht1_m : ∀ x ∈ p1, f1 x ∈ s x
ht1_eq : t1 = ⋂ x ∈ p1, f1 x
p2 : Finset ι
hp2 : ↑p2 ⊆ T
f2 : ι → Set Ω
ht2_m : ∀ x ∈ p2, f2 x ∈ s x
ht2_eq : t2 = ⋂ x ∈ p2, f2 x
g : ι → Set Ω := fun i ↦ (if i ∈ p1 then f1 i else univ) ∩ if i ∈ p2 then f2 i else univ
hgm : ∀ i ∈ p1 ∪ p2, g i ∈ s i
h_p1_inter_p2 :
  (⋂ x ∈ p1, f1 x) ∩ ⋂ x ∈ p2, f2 x = ⋂ i ∈ p1 ∪ p2, (if i ∈ p1 then f1 i else univ) ∩ if i ∈ p2 then f2 i else univ
a : α
ha : (κ a) (⋂ i ∈ p1 ∪ p2, g i) = ∏ i ∈ p1 ∪ p2, (κ a) (g i)
⊢ (κ a) ((⋂ x ∈ p1, f1 x) ∩ ⋂ x ∈ p2, f2 x) = ∏ n ∈ p1 ∪ p2, (κ a) (g n)

Tactic:
h_p1_inter_p2,

Goal state after:
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
s : ι → Set (Set Ω)
S T : Set ι
h_indep : iIndepSets s κ μ
hST : Disjoint S T
t1 t2 : Set Ω
p1 : Finset ι
hp1 : ↑p1 ⊆ S
f1 : ι → Set Ω
ht1_m : ∀ x ∈ p1, f1 x ∈ s x
ht1_eq : t1 = ⋂ x ∈ p1, f1 x
p2 : Finset ι
hp2 : ↑p2 ⊆ T
f2 : ι → Set Ω
ht2_m : ∀ x ∈ p2, f2 x ∈ s x
ht2_eq : t2 = ⋂ x ∈ p2, f2 x
g : ι → Set Ω := fun i ↦ (if i ∈ p1 then f1 i else univ) ∩ if i ∈ p2 then f2 i else univ
hgm : ∀ i ∈ p1 ∪ p2, g i ∈ s i
h_p1_inter_p2 :
  (⋂ x ∈ p1, f1 x) ∩ ⋂ x ∈ p2, f2 x = ⋂ i ∈ p1 ∪ p2, (if i ∈ p1 then f1 i else univ) ∩ if i ∈ p2 then f2 i else univ
a : α
ha : (κ a) (⋂ i ∈ p1 ∪ p2, g i) = ∏ i ∈ p1 ∪ p2, (κ a) (g i)
⊢ (κ a) (⋂ i ∈ p1 ∪ p2, (if i ∈ p1 then f1 i else univ) ∩ if i ∈ p2 then f2 i else univ) = ∏ n ∈ p1 ∪ p2, (κ a) (g n)