mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case h.refine_3.left
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := ⋯
y : Fin N → ℝ := ⋯
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := ⋯
n✝ : Fin N
x✝ : X
hx : (a + ε * (↑↑n✝ + 1) - ε < f x✝ ∧ f x✝ ≤ a + ε * (↑↑n✝ + 1)) ∧ x✝ ∈ tsupport ⇑f
⊢ a + ε * ↑↑n✝ < f x✝

pp: case h.refine_3.right
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := ⋯
y : Fin N → ℝ := ⋯
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := ⋯
n✝ : Fin N
x✝ : X
hx : (a + ε * (↑↑n✝ + 1) - ε < f x✝ ∧ f x✝ ≤ a + ε * (↑↑n✝ + 1)) ∧ x✝ ∈ tsupport ⇑f
⊢ f x✝ ≤ a + ε * (↑↑n✝ + 1)

Tactic:
<;>

Goal state after:
case h.refine_3.left
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := ⋯
y : Fin N → ℝ := ⋯
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := ⋯
n✝ : Fin N
x✝ : X
hx : (a + ε * (↑↑n✝ + 1) - ε < f x✝ ∧ f x✝ ≤ a + ε * (↑↑n✝ + 1)) ∧ x✝ ∈ tsupport ⇑f
⊢ a + ε * ↑↑n✝ < f x✝

case h.refine_3.right
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := ⋯
y : Fin N → ℝ := ⋯
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := ⋯
n✝ : Fin N
x✝ : X
hx : (a + ε * (↑↑n✝ + 1) - ε < f x✝ ∧ f x✝ ≤ a + ε * (↑↑n✝ + 1)) ∧ x✝ ∈ tsupport ⇑f
⊢ f x✝ ≤ a + ε * (↑↑n✝ + 1)
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
h0 : ∫ (x : Ω), X x ∂P = 0
h1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1
t : ℝ
⊢ 1 + ↑(∫ (x : Ω), X x ∂P) * ↑t * I - ↑(∫ (x : Ω), (X ^ 2) x ∂P) * ↑t ^ 2 / 2 = 1 - ↑t ^ 2 / 2

pp: case hint
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
h0 : ∫ (x : Ω), X x ∂P = 0
h1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1
t : ℝ
⊢ MemLp id 2 (Measure.map X P)

Tactic:
h0,

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
h0 : ∫ (x : Ω), X x ∂P = 0
h1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1
t : ℝ
⊢ 1 + ↑0 * ↑t * I - ↑(∫ (x : Ω), (X ^ 2) x ∂P) * ↑t ^ 2 / 2 = 1 - ↑t ^ 2 / 2

case hint
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
h0 : ∫ (x : Ω), X x ∂P = 0
h1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1
t : ℝ
⊢ MemLp id 2 (Measure.map X P)
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case hbc
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
μ : Measure X := ⋯
K : Set X := ⋯
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := ⋯
hε' : 0 < ε' ∧ ε' * (2 * μ.real K + |a| + b + ε') ≤ ε
E : Fin N → Set X
hE :
  tsupport ⇑f = ⋃ j, E j ∧
    univ.PairwiseDisjoint E ∧
      (∀ (n : Fin N), ∀ x ∈ E n, a + ε' * ↑↑n < f x ∧ f x ≤ a + ε' * (↑↑n + 1)) ∧ ∀ (n : Fin N), MeasurableSet (E n)
y : Fin N → ℝ := ⋯
hE' : ∀ (n : Fin N), μ (E n) ≠ ∞
V : Fin N → Opens X
hV : ∀ (n : Fin N), E n ⊆ ↑(V n) ∧ (∀ x ∈ V n, f x < y n + ε') ∧ μ ↑(V n) ≤ μ (E n) + ENNReal.ofReal (ε' / ↑N)
g : Fin N → X →C_c ℝ
hg :
  (∀ (n : Fin N), tsupport ⇑(g n) ⊆ (V n).carrier) ∧
    EqOn (∑ n, ⇑(g n)) 1 (tsupport f.toFun) ∧
      (∀ (n : Fin N) (x : X), (g n) x ∈ Icc 0 1) ∧ ∀ (n : Fin N), HasCompactSupport ⇑(g n)
⊢ μ.real K ≤ ∑ n, Λ (g n)

Tactic:
← map_sum Λ g _

Goal state after:
case hbc
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
μ : Measure X := rieszMeasure Λ
K : Set X := tsupport ⇑f
ε : ℝ
hε : 0 < ε
a b : ℝ
hab : a < b ∧ range ⇑f ⊆ Ioo a b
N : ℕ
hN : 0 < N
ε' : ℝ := (b - a) / ↑N
hε' : 0 < ε' ∧ ε' * (2 * μ.real K + |a| + b + ε') ≤ ε
E : Fin N → Set X
hE :
  tsupport ⇑f = ⋃ j, E j ∧
    univ.PairwiseDisjoint E ∧
      (∀ (n : Fin N), ∀ x ∈ E n, a + ε' * ↑↑n < f x ∧ f x ≤ a + ε' * (↑↑n + 1)) ∧ ∀ (n : Fin N), MeasurableSet (E n)
y : Fin N → ℝ := fun n ↦ a + ε' * (↑↑n + 1)
hE' : ∀ (n : Fin N), μ (E n) ≠ ∞
V : Fin N → Opens X
hV : ∀ (n : Fin N), E n ⊆ ↑(V n) ∧ (∀ x ∈ V n, f x < y n + ε') ∧ μ ↑(V n) ≤ μ (E n) + ENNReal.ofReal (ε' / ↑N)
g : Fin N → X →C_c ℝ
hg :
  (∀ (n : Fin N), tsupport ⇑(g n) ⊆ (V n).carrier) ∧
    EqOn (∑ n, ⇑(g n)) 1 (tsupport f.toFun) ∧
      (∀ (n : Fin N) (x : X), (g n) x ∈ Icc 0 1) ∧ ∀ (n : Fin N), HasCompactSupport ⇑(g n)
⊢ μ.real K ≤ Λ (∑ x, g x)
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: case a
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
x : X
U : Set X
hxU : x ∈ U
hU : IsOpen[inst✝¹] U
f : X → ↑I
hf : Continuous[inst✝¹, _] f
efx : f x = 0
hfU : stoneCechUnit ⁻¹' stoneCechExtend hf ⁻¹' {1}ᶜ ⊆ U
⊢ ∃ i, stoneCechUnit x ∈ i ∧ IsOpen i ∧ stoneCechUnit ⁻¹' i ⊆ U

Tactic:
refine ⟨stoneCechExtend hf ⁻¹' {1}ᶜ, ?_,
      isOpen_compl_singleton.preimage (continuous_stoneCechExtend hf), hfU⟩

Goal state after:
case a
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
x : X
U : Set X
hxU : x ∈ U
hU : IsOpen[inst✝¹] U
f : X → ↑I
hf : Continuous[inst✝¹, _] f
efx : f x = 0
hfU : stoneCechUnit ⁻¹' stoneCechExtend hf ⁻¹' {1}ᶜ ⊆ U
⊢ stoneCechUnit x ∈ stoneCechExtend hf ⁻¹' {1}ᶜ
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
mF : MeasurableSpace F
μ : Measure E
ν : Measure F
p : ℝ≥0∞
inst✝² : Fact (1 ≤ p)
inst✝¹ : SFinite μ
inst✝ : SFinite ν
L : StrongDual ℝ (WithLp p (E × F))
⊢ ∫ (v : WithLp p (E × F)), cexp (↑(L v) * I) ∂Measure.map (toLp p) (μ.prod ν) =
    ∫ (z : E × F),
      cexp (↑((L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inl ℝ E F) z.1) * I) *
        cexp (↑((L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inr ℝ E F) z.2) * I) ∂μ.prod ν

Tactic:
← Complex.exp_add,

Goal state after:
E : Type u_2
F : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
mF : MeasurableSpace F
μ : Measure E
ν : Measure F
p : ℝ≥0∞
inst✝² : Fact (1 ≤ p)
inst✝¹ : SFinite μ
inst✝ : SFinite ν
L : StrongDual ℝ (WithLp p (E × F))
⊢ ∫ (v : WithLp p (E × F)), cexp (↑(L v) * I) ∂Measure.map (toLp p) (μ.prod ν) =
    ∫ (z : E × F),
      cexp
        (↑((L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inl ℝ E F) z.1) * I +
          ↑((L ∘SL ↑(prodContinuousLinearEquiv p ℝ E F).symm ∘SL ContinuousLinearMap.inr ℝ E F) z.2) * I) ∂μ.prod ν
Module: Mathlib.MeasureTheory.Measure.DiracProba

Goal state before:
pp: X : Type u_1
inst✝² : MeasurableSpace X
inst✝¹ : TopologicalSpace X
inst✝ : OpensMeasurableSpace X
⊢ ∀ (x : X), ContinuousAt (fun x ↦ diracProba x) x

Tactic:
apply fun x ↦ ProbabilityMeasure.tendsto_iff_forall_lintegral_tendsto.mpr fun f ↦ ?_

Goal state after:
X : Type u_1
inst✝² : MeasurableSpace X
inst✝¹ : TopologicalSpace X
inst✝ : OpensMeasurableSpace X
x : X
f : X →ᵇ ℝ≥0
⊢ Tendsto (fun i ↦ ∫⁻ (ω : X), ↑(f ω) ∂↑(diracProba i)) (𝓝 x) (𝓝 (∫⁻ (ω : X), ↑(f ω) ∂↑(diracProba x)))
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: p : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ p)
ι : Type u_4
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
E : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace ℝ (E i)
mE : (i : ι) → MeasurableSpace (E i)
μ : (i : ι) → Measure (E i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
L : StrongDual ℝ (PiLp p E)
⊢ ∫ (v : WithLp p ((i : ι) → E i)), cexp (↑(L v) * I) ∂Measure.map (toLp p) (Measure.pi μ) =
    ∫ (x : (i : ι) → E i),
      cexp
        (↑(∑ x_1, L ((↑(PiLp.continuousLinearEquiv p ℝ E).symm ∘SL ContinuousLinearMap.single ℝ E x_1) (x x_1))) *
          I) ∂Measure.pi μ

Tactic:
← map_sum,

Goal state after:
p : ℝ≥0∞
inst✝⁵ : Fact (1 ≤ p)
ι : Type u_4
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
E : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace ℝ (E i)
mE : (i : ι) → MeasurableSpace (E i)
μ : (i : ι) → Measure (E i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
L : StrongDual ℝ (PiLp p E)
⊢ ∫ (v : WithLp p ((i : ι) → E i)), cexp (↑(L v) * I) ∂Measure.map (toLp p) (Measure.pi μ) =
    ∫ (x : (i : ι) → E i),
      cexp
        (↑(L (∑ x_1, (↑(PiLp.continuousLinearEquiv p ℝ E).symm ∘SL ContinuousLinearMap.single ℝ E x_1) (x x_1))) *
          I) ∂Measure.pi μ
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: case refine_1
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ ν : Measure Ω
B : Set Ω
x✝ : MeasurableSet B
⊢ μ B ≤ ν (thickening (max (μ univ) (ν univ)).toReal B) + max (μ univ) (ν univ)

pp: case refine_2
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ ν : Measure Ω
B : Set Ω
x✝ : MeasurableSet B
⊢ ν B ≤ μ (thickening (max (μ univ) (ν univ)).toReal B) + max (μ univ) (ν univ)

Tactic:
<;>

Goal state after:
case refine_1
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ ν : Measure Ω
B : Set Ω
x✝ : MeasurableSet B
⊢ μ B ≤ ν (thickening (max (μ univ) (ν univ)).toReal B) + max (μ univ) (ν univ)

case refine_2
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ ν : Measure Ω
B : Set Ω
x✝ : MeasurableSet B
⊢ ν B ≤ μ (thickening (max (μ univ) (ν univ)).toReal B) + max (μ univ) (ν univ)
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: case refine_1
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ ν : Measure Ω
B : Set Ω
x✝ : MeasurableSet B
⊢ μ B ≤ max (μ univ) (ν univ)

pp: case refine_2
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ ν : Measure Ω
B : Set Ω
x✝ : MeasurableSet B
⊢ ν B ≤ max (μ univ) (ν univ)

Tactic:
<;>

Goal state after:
case refine_1
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ ν : Measure Ω
B : Set Ω
x✝ : MeasurableSet B
⊢ μ B ≤ max (μ univ) (ν univ)

case refine_2
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ ν : Measure Ω
B : Set Ω
x✝ : MeasurableSet B
⊢ ν B ≤ max (μ univ) (ν univ)
Module: Mathlib.MeasureTheory.Measure.Haar.DistribChar

Goal state before:
pp: case hb₀
G : Type u_1
A : Type u_2
inst✝¹⁰ : Group G
inst✝⁹ : AddCommGroup A
inst✝⁸ : DistribMulAction G A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : IsTopologicalAddGroup A
inst✝⁵ : LocallyCompactSpace A
inst✝⁴ : ContinuousConstSMul G A
inst✝³ : MeasurableSpace A
inst✝² : BorelSpace A
μ : Measure A
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : μ.Regular
s : Set A
hs₀ : μ s ≠ 0
hs : μ s ≠ ∞
g : G
⊢ μ s ≠ 0

pp: case hb
G : Type u_1
A : Type u_2
inst✝¹⁰ : Group G
inst✝⁹ : AddCommGroup A
inst✝⁸ : DistribMulAction G A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : IsTopologicalAddGroup A
inst✝⁵ : LocallyCompactSpace A
inst✝⁴ : ContinuousConstSMul G A
inst✝³ : MeasurableSpace A
inst✝² : BorelSpace A
μ : Measure A
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : μ.Regular
s : Set A
hs₀ : μ s ≠ 0
hs : μ s ≠ ∞
g : G
⊢ μ s ≠ ∞

Tactic:
<;>

Goal state after:
case hb₀
G : Type u_1
A : Type u_2
inst✝¹⁰ : Group G
inst✝⁹ : AddCommGroup A
inst✝⁸ : DistribMulAction G A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : IsTopologicalAddGroup A
inst✝⁵ : LocallyCompactSpace A
inst✝⁴ : ContinuousConstSMul G A
inst✝³ : MeasurableSpace A
inst✝² : BorelSpace A
μ : Measure A
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : μ.Regular
s : Set A
hs₀ : μ s ≠ 0
hs : μ s ≠ ∞
g : G
⊢ μ s ≠ 0

case hb
G : Type u_1
A : Type u_2
inst✝¹⁰ : Group G
inst✝⁹ : AddCommGroup A
inst✝⁸ : DistribMulAction G A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : IsTopologicalAddGroup A
inst✝⁵ : LocallyCompactSpace A
inst✝⁴ : ContinuousConstSMul G A
inst✝³ : MeasurableSpace A
inst✝² : BorelSpace A
μ : Measure A
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : μ.Regular
s : Set A
hs₀ : μ s ≠ 0
hs : μ s ≠ ∞
g : G
⊢ μ s ≠ ∞
Module: Mathlib.MeasureTheory.Measure.Haar.DistribChar

Goal state before:
pp: case hb₀
G : Type u_1
A : Type u_2
inst✝¹⁰ : Group G
inst✝⁹ : AddCommGroup A
inst✝⁸ : DistribMulAction G A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : IsTopologicalAddGroup A
inst✝⁵ : LocallyCompactSpace A
inst✝⁴ : ContinuousConstSMul G A
g : G
inst✝³ : MeasurableSpace A
inst✝² : BorelSpace A
μ : Measure A
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : μ.Regular
s : Set A
hs₀ : μ s ≠ 0
hs : μ s ≠ ∞
r : ℝ≥0
hμgs : μ (g • s) = ↑r * μ s
⊢ μ s ≠ 0

pp: case hb
G : Type u_1
A : Type u_2
inst✝¹⁰ : Group G
inst✝⁹ : AddCommGroup A
inst✝⁸ : DistribMulAction G A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : IsTopologicalAddGroup A
inst✝⁵ : LocallyCompactSpace A
inst✝⁴ : ContinuousConstSMul G A
g : G
inst✝³ : MeasurableSpace A
inst✝² : BorelSpace A
μ : Measure A
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : μ.Regular
s : Set A
hs₀ : μ s ≠ 0
hs : μ s ≠ ∞
r : ℝ≥0
hμgs : μ (g • s) = ↑r * μ s
⊢ μ s ≠ ∞

Tactic:
<;>

Goal state after:
case hb₀
G : Type u_1
A : Type u_2
inst✝¹⁰ : Group G
inst✝⁹ : AddCommGroup A
inst✝⁸ : DistribMulAction G A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : IsTopologicalAddGroup A
inst✝⁵ : LocallyCompactSpace A
inst✝⁴ : ContinuousConstSMul G A
g : G
inst✝³ : MeasurableSpace A
inst✝² : BorelSpace A
μ : Measure A
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : μ.Regular
s : Set A
hs₀ : μ s ≠ 0
hs : μ s ≠ ∞
r : ℝ≥0
hμgs : μ (g • s) = ↑r * μ s
⊢ μ s ≠ 0

case hb
G : Type u_1
A : Type u_2
inst✝¹⁰ : Group G
inst✝⁹ : AddCommGroup A
inst✝⁸ : DistribMulAction G A
inst✝⁷ : TopologicalSpace A
inst✝⁶ : IsTopologicalAddGroup A
inst✝⁵ : LocallyCompactSpace A
inst✝⁴ : ContinuousConstSMul G A
g : G
inst✝³ : MeasurableSpace A
inst✝² : BorelSpace A
μ : Measure A
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : μ.Regular
s : Set A
hs₀ : μ s ≠ 0
hs : μ s ≠ ∞
r : ℝ≥0
hμgs : μ (g • s) = ↑r * μ s
⊢ μ s ≠ ∞
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this : (ENNReal.ofReal p).toReal = p
⊢ volume {x | ∑ i, ‖x i‖ ^ p < 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))

Tactic:
have h₂ : ∀ x : ι → ℂ, 0 ≤ ∑ i, ‖x i‖ ^ p := by
    refine fun _ => Finset.sum_nonneg' ?_
    exact fun i => (fun _ => rpow_nonneg (norm_nonneg _) _) _

Goal state after:
ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
this : (ENNReal.ofReal p).toReal = p
h₂ : ∀ (x : ι → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p
⊢ volume {x | ∑ i, ‖x i‖ ^ p < 1} =
    ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
g : E →ᵇ ℝ
⊢ Tendsto (fun i ↦ ∫ (x : E), g x ∂↑(id i)) (↑f) (𝓝 (∫ (x : E), g x ∂↑μ))

Tactic:
rw [Metric.tendsto_nhds]

Goal state after:
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
h : NormalSpace E ∨ Monotone K
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
g : E →ᵇ ℝ
⊢ ∀ ε > 0, ∀ᶠ (x : FiniteMeasure E) in ↑f, dist (∫ (x : E), g x ∂↑(id x)) (∫ (x : E), g x ∂↑μ) < ε
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_2
ι : Type u_3
inst✝³ : RCLike 𝕜
inst✝² : Fintype ι
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
b : OrthonormalBasis ι 𝕜 E
h : ∀ (i : ι), Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖⟪b i, x⟫_𝕜‖}) atTop (𝓝 0)
hE : Nontrivial E
h_rank : 0 < ↑(Fintype.card ι)
this✝ : Nonempty ι
this : ProperSpace E
r : ℝ
⊢ ⨆ μ ∈ S, ∑ i, μ {x | r / √↑(Fintype.card ι) < ‖⟪b i, x⟫_𝕜‖} ≤
    ∑ i, ⨆ μ ∈ S, μ {x | r / √↑(Fintype.card ι) < ‖⟪b i, x⟫_𝕜‖}

Tactic:
refine iSup_le fun μ ↦ (iSup_le fun hμS ↦ ?_)
      gcongr with i
      exact le_biSup (fun μ ↦ μ {x | r / √(Fintype.card ι) < ‖⟪b i, x⟫_𝕜‖}) hμS

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

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 : ∀ (n : ℕ) (r : ℝ), 0 < r → 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ),
    0 < r → (μ n).real {x | r < |⟪z, x⟫|} ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
⊢ Tendsto (fun n ↦ ∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 1 - charFun (μ n) (t • z)) atTop
    (𝓝 (∫ (t : ℝ) in -(2 * r⁻¹)..2 * r⁻¹, 1 - f (t • z)))

Tactic:
simp_rw [intervalIntegral.integral_of_le hr']

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 : ∀ (n : ℕ) (r : ℝ), 0 < r → 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ),
    0 < r → (μ n).real {x | r < |⟪z, x⟫|} ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
⊢ Tendsto (fun n ↦ ∫ (t : ℝ) in Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹), 1 - charFun (μ n) (t • z) ∂volume) atTop
    (𝓝 (∫ (t : ℝ) in Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹), 1 - f (t • z) ∂volume))
Module: Mathlib.MeasureTheory.Measure.PreVariation

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s : ℕ → Set X
hs : ∀ (i : ℕ), MeasurableSet (s i)
hs' : Pairwise (Disjoint on s)
P : (i : ℕ) → Finpartition ⟨s i, ⋯⟩
n : ℕ
s' : ℕ → Subtype MeasurableSet := fun i ↦ ⟨s i, ⋯⟩
hs_disj : (↑(Finset.range n)).PairwiseDisjoint s'
Q : Finpartition ((Finset.range n).sup fun i ↦ ⟨s i, ⋯⟩) := Finpartition.combine P ⋯
⊢ (Finset.range n).sup s' ≤ ⟨⋃ i, s i, ⋯⟩

Tactic:
← Subtype.coe_le_coe,

Goal state after:
X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s : ℕ → Set X
hs : ∀ (i : ℕ), MeasurableSet (s i)
hs' : Pairwise (Disjoint on s)
P : (i : ℕ) → Finpartition ⟨s i, ⋯⟩
n : ℕ
s' : ℕ → Subtype MeasurableSet := fun i ↦ ⟨s i, ⋯⟩
hs_disj : (↑(Finset.range n)).PairwiseDisjoint s'
Q : Finpartition ((Finset.range n).sup fun i ↦ ⟨s i, ⋯⟩) := Finpartition.combine P ⋯
⊢ ↑((Finset.range n).sup s') ⊆ ↑⟨⋃ i, s i, ⋯⟩
Module: Mathlib.MeasureTheory.Measure.HasOuterApproxClosedProd

Goal state before:
pp: case e'_2
ι : Type u_1
T : Type u_4
X : ι → Type u_5
mX : (i : ι) → MeasurableSpace (X i)
inst✝⁸ : (i : ι) → TopologicalSpace (X i)
inst✝⁷ : ∀ (i : ι), BorelSpace (X i)
inst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)
mT : MeasurableSpace T
inst✝⁵ : TopologicalSpace T
inst✝⁴ : BorelSpace T
inst✝³ : HasOuterApproxClosed T
inst✝² : Finite ι
μ ν : Measure (((i : ι) → X i) × T)
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h :
  ∀ (f : ((i : ι) → X i) →ᵇ ℝ) (g : T →ᵇ ℝ),
    ∫ (p : ((i : ι) → X i) × T), f p.1 * g p.2 ∂μ = ∫ (p : ((i : ι) → X i) × T), f p.1 * g p.2 ∂ν
this : Fintype ι
f : (i : ι) → X i →ᵇ ℝ
g : T →ᵇ ℝ
x✝ : ((i : ι) → X i) × T
⊢ ∏ i, (f i) (x✝.1 i) = (∏ i, (f i).compContinuous { toFun := Function.eval i, continuous_toFun := ⋯ }) x✝.1

pp: case e'_3
ι : Type u_1
T : Type u_4
X : ι → Type u_5
mX : (i : ι) → MeasurableSpace (X i)
inst✝⁸ : (i : ι) → TopologicalSpace (X i)
inst✝⁷ : ∀ (i : ι), BorelSpace (X i)
inst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)
mT : MeasurableSpace T
inst✝⁵ : TopologicalSpace T
inst✝⁴ : BorelSpace T
inst✝³ : HasOuterApproxClosed T
inst✝² : Finite ι
μ ν : Measure (((i : ι) → X i) × T)
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h :
  ∀ (f : ((i : ι) → X i) →ᵇ ℝ) (g : T →ᵇ ℝ),
    ∫ (p : ((i : ι) → X i) × T), f p.1 * g p.2 ∂μ = ∫ (p : ((i : ι) → X i) × T), f p.1 * g p.2 ∂ν
this : Fintype ι
f : (i : ι) → X i →ᵇ ℝ
g : T →ᵇ ℝ
x✝ : ((i : ι) → X i) × T
⊢ ∏ i, (f i) (x✝.1 i) = (∏ i, (f i).compContinuous { toFun := Function.eval i, continuous_toFun := ⋯ }) x✝.1

Tactic:
<;>

Goal state after:
case e'_2
ι : Type u_1
T : Type u_4
X : ι → Type u_5
mX : (i : ι) → MeasurableSpace (X i)
inst✝⁸ : (i : ι) → TopologicalSpace (X i)
inst✝⁷ : ∀ (i : ι), BorelSpace (X i)
inst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)
mT : MeasurableSpace T
inst✝⁵ : TopologicalSpace T
inst✝⁴ : BorelSpace T
inst✝³ : HasOuterApproxClosed T
inst✝² : Finite ι
μ ν : Measure (((i : ι) → X i) × T)
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h :
  ∀ (f : ((i : ι) → X i) →ᵇ ℝ) (g : T →ᵇ ℝ),
    ∫ (p : ((i : ι) → X i) × T), f p.1 * g p.2 ∂μ = ∫ (p : ((i : ι) → X i) × T), f p.1 * g p.2 ∂ν
this : Fintype ι
f : (i : ι) → X i →ᵇ ℝ
g : T →ᵇ ℝ
x✝ : ((i : ι) → X i) × T
⊢ ∏ i, (f i) (x✝.1 i) = (∏ i, (f i).compContinuous { toFun := Function.eval i, continuous_toFun := ⋯ }) x✝.1

case e'_3
ι : Type u_1
T : Type u_4
X : ι → Type u_5
mX : (i : ι) → MeasurableSpace (X i)
inst✝⁸ : (i : ι) → TopologicalSpace (X i)
inst✝⁷ : ∀ (i : ι), BorelSpace (X i)
inst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)
mT : MeasurableSpace T
inst✝⁵ : TopologicalSpace T
inst✝⁴ : BorelSpace T
inst✝³ : HasOuterApproxClosed T
inst✝² : Finite ι
μ ν : Measure (((i : ι) → X i) × T)
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h :
  ∀ (f : ((i : ι) → X i) →ᵇ ℝ) (g : T →ᵇ ℝ),
    ∫ (p : ((i : ι) → X i) × T), f p.1 * g p.2 ∂μ = ∫ (p : ((i : ι) → X i) × T), f p.1 * g p.2 ∂ν
this : Fintype ι
f : (i : ι) → X i →ᵇ ℝ
g : T →ᵇ ℝ
x✝ : ((i : ι) → X i) × T
⊢ ∏ i, (f i) (x✝.1 i) = (∏ i, (f i).compContinuous { toFun := Function.eval i, continuous_toFun := ⋯ }) x✝.1