mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.UnifTight

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : NormedAddCommGroup β
x✝ : MeasurableSpace α
f : ι → α → β
p : ℝ≥0∞
μ : Measure α
⊢ UnifTight f p μ ↔ ∀ ⦃ε : ℝ≥0∞⦄, 0 < ε → ∃ s, μ s ≠ ∞ ∧ ∀ (i : ι), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ε

Tactic:
by
  simp only [ENNReal.forall_ennreal, ENNReal.coe_pos]
  refine (and_iff_left ?_).symm
  simp only [zero_lt_top, le_top, implies_true, and_true, true_implies]
  use ∅; simpa only [measure_empty] using zero_ne_top

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

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifTight f p μ
hg : UnifTight g p μ
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
ε : ℝ≥0
hε : 0 < ε
η : ℝ≥0∞
hη_pos : 0 < η
hη :
  ∀ (f g : α → β),
    AEStronglyMeasurable f μ →
      AEStronglyMeasurable g μ → eLpNorm f p μ ≤ η → eLpNorm g p μ ≤ η → eLpNorm (f + g) p μ < ↑ε
hη_top : ¬η = ∞
⊢ ∃ s, μ s ≠ ∞ ∧ ∀ (i : ι), eLpNorm (sᶜ.indicator ((f + g) i)) p μ ≤ ↑ε

Tactic:
obtain ⟨s, hμs, hsm, hfs, hgs⟩ :
      ∃ s ∈ μ.cofinite, MeasurableSet s ∧
        (∀ i, eLpNorm (s.indicator (f i)) p μ ≤ η) ∧
        (∀ i, eLpNorm (s.indicator (g i)) p μ ≤ η) :=
    ((hf.eventually_cofinite_indicator hη_pos.ne').and
      (hg.eventually_cofinite_indicator hη_pos.ne')).exists_measurable_mem_of_smallSets

Goal state after:
case neg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifTight f p μ
hg : UnifTight g p μ
hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ
ε : ℝ≥0
hε : 0 < ε
η : ℝ≥0∞
hη_pos : 0 < η
hη :
  ∀ (f g : α → β),
    AEStronglyMeasurable f μ →
      AEStronglyMeasurable g μ → eLpNorm f p μ ≤ η → eLpNorm g p μ ≤ η → eLpNorm (f + g) p μ < ↑ε
hη_top : ¬η = ∞
s : Set α
hμs : s ∈ μ.cofinite
hsm : MeasurableSet s
hfs : ∀ (i : ι), eLpNorm (s.indicator (f i)) p μ ≤ η
hgs : ∀ (i : ι), eLpNorm (s.indicator (g i)) p μ ≤ η
⊢ ∃ s, μ s ≠ ∞ ∧ ∀ (i : ι), eLpNorm (sᶜ.indicator ((f + g) i)) p μ ≤ ↑ε
Module: Mathlib.MeasureTheory.Group.GeometryOfNumbers

Goal state before:
pp: E : Type u_1
inst✝⁸ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : Nontrivial E
inst✝² : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝¹ : Countable ↥L
inst✝ : DiscreteTopology ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h_cpt : IsCompact s
h : μ F * 2 ^ finrank ℝ E ≤ μ s
h_mes : μ s ≠ 0
h_nemp : s.Nonempty
u : ℕ → ℝ≥0 := ⋯.choose
K : ConvexBody E := { carrier := s, convex' := h_conv, isCompact' := h_cpt, nonempty' := h_nemp }
S : ℕ → ConvexBody E := fun n ↦ (1 + u n) • K
Z : ℕ → Set E := fun n ↦ ↑(S n) ∩ (↑L \ {0})
⊢ ∃ x, x ≠ 0 ∧ ↑x ∈ s

Tactic:
have h_zero : 0 ∈ K := K.zero_mem_of_symmetric h_symm

Goal state after:
E : Type u_1
inst✝⁸ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : Nontrivial E
inst✝² : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝¹ : Countable ↥L
inst✝ : DiscreteTopology ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h_cpt : IsCompact s
h : μ F * 2 ^ finrank ℝ E ≤ μ s
h_mes : μ s ≠ 0
h_nemp : s.Nonempty
u : ℕ → ℝ≥0 := ⋯.choose
K : ConvexBody E := { carrier := s, convex' := h_conv, isCompact' := h_cpt, nonempty' := h_nemp }
S : ℕ → ConvexBody E := fun n ↦ (1 + u n) • K
Z : ℕ → Set E := fun n ↦ ↑(S n) ∩ (↑L \ {0})
h_zero : 0 ∈ K
⊢ ∃ x, x ≠ 0 ∧ ↑x ∈ s
Module: Mathlib.MeasureTheory.Integral.CircleTransform

Goal state before:
pp: R r : ℝ
hr : r < R
z : ℂ
⊢ ContinuousOn ((fun x ↦ ‖x‖) ∘ circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ)

Tactic:
by
  have : ContinuousOn (circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ) := by
    apply_rules [ContinuousOn.fun_smul, continuousOn_const]
    · simp only [deriv_circleMap]
      apply_rules [ContinuousOn.mul, (continuous_circleMap 0 R).comp_continuousOn continuousOn_snd,
        continuousOn_const]
    · simpa only [inv_pow] using continuousOn_prod_circle_transform_function hr
  exact this.norm

Goal state after:
Module: Mathlib.MeasureTheory.Group.GeometryOfNumbers

Goal state before:
pp: case refine_1
E : Type u_1
inst✝⁸ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : Nontrivial E
inst✝² : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝¹ : Countable ↥L
inst✝ : DiscreteTopology ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h_cpt : IsCompact s
h : μ F * 2 ^ finrank ℝ E ≤ μ s
h_mes : μ s ≠ 0
h_nemp : s.Nonempty
u : ℕ → ℝ≥0 := ⋯.choose
K : ConvexBody E := { carrier := s, convex' := h_conv, isCompact' := h_cpt, nonempty' := h_nemp }
S : ℕ → ConvexBody E := fun n ↦ (1 + u n) • K
Z : ℕ → Set E := fun n ↦ ↑(S n) ∩ (↑L \ {0})
h_zero : 0 ∈ K
h_clos : IsClosed (↑L \ {0})
n : ℕ
⊢ Z (n + 1) ⊆ Z n

pp: case refine_2
E : Type u_1
inst✝⁸ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : Nontrivial E
inst✝² : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝¹ : Countable ↥L
inst✝ : DiscreteTopology ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h_cpt : IsCompact s
h : μ F * 2 ^ finrank ℝ E ≤ μ s
h_mes : μ s ≠ 0
h_nemp : s.Nonempty
u : ℕ → ℝ≥0 := ⋯.choose
K : ConvexBody E := { carrier := s, convex' := h_conv, isCompact' := h_cpt, nonempty' := h_nemp }
S : ℕ → ConvexBody E := fun n ↦ (1 + u n) • K
Z : ℕ → Set E := fun n ↦ ↑(S n) ∩ (↑L \ {0})
h_zero : 0 ∈ K
h_clos : IsClosed (↑L \ {0})
n : ℕ
⊢ (Z n).Nonempty

Tactic:
· refine Set.inter_subset_inter_left _ (SetLike.coe_subset_coe.mpr ?_)
    refine ConvexBody.smul_le_of_le K h_zero ?_
    rw [add_le_add_iff_left]
    exact le_of_lt <| (exists_seq_strictAnti_tendsto (0 : ℝ≥0)).choose_spec.1 (Nat.lt_add_one n)

Goal state after:
case refine_2
E : Type u_1
inst✝⁸ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : Nontrivial E
inst✝² : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝¹ : Countable ↥L
inst✝ : DiscreteTopology ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h_cpt : IsCompact s
h : μ F * 2 ^ finrank ℝ E ≤ μ s
h_mes : μ s ≠ 0
h_nemp : s.Nonempty
u : ℕ → ℝ≥0 := ⋯.choose
K : ConvexBody E := { carrier := s, convex' := h_conv, isCompact' := h_cpt, nonempty' := h_nemp }
S : ℕ → ConvexBody E := fun n ↦ (1 + u n) • K
Z : ℕ → Set E := fun n ↦ ↑(S n) ∩ (↑L \ {0})
h_zero : 0 ∈ K
h_clos : IsClosed (↑L \ {0})
n : ℕ
⊢ (Z n).Nonempty
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
a : E
r : ℝ
hr : 0 ≤ r
hdf : ∀ᶠ (x : E) in 𝓝 a, DifferentiableAt ℝ f x
hderiv : fderiv ℝ f =O[𝓝 a] fun x ↦ ‖x - a‖ ^ r
hf₀ : f a = 0
⊢ f =O[𝓝 a] fun x ↦ ‖x - a‖ ^ (r + 1)

Tactic:
by
  simpa [hf₀] using sub_isBigO_norm_rpow_add_one_of_fderiv hr hdf hderiv

Goal state after:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K1 K2 : Compacts X
ε : ℝ≥0
εpos : 0 < ε
f1 : X →C_c ℝ≥0
f_test_function_K1 : (∀ x ∈ K1, 1 ≤ f1 x) ∧ Λ f1 < rieszContentAux Λ K1 + ε / 2
f2 : X →C_c ℝ≥0
f_test_function_K2 : (∀ x ∈ K2, 1 ≤ f2 x) ∧ Λ f2 < rieszContentAux Λ K2 + ε / 2
f_test_function_union : ∀ x ∈ K1 ⊔ K2, 1 ≤ (f1 + f2) x
⊢ rieszContentAux Λ K1 + (rieszContentAux Λ K2 + (ε / 2 + ε / 2)) = rieszContentAux Λ K1 + rieszContentAux Λ K2 + ε

Tactic:
add_halves ε,

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K1 K2 : Compacts X
ε : ℝ≥0
εpos : 0 < ε
f1 : X →C_c ℝ≥0
f_test_function_K1 : (∀ x ∈ K1, 1 ≤ f1 x) ∧ Λ f1 < rieszContentAux Λ K1 + ε / 2
f2 : X →C_c ℝ≥0
f_test_function_K2 : (∀ x ∈ K2, 1 ≤ f2 x) ∧ Λ f2 < rieszContentAux Λ K2 + ε / 2
f_test_function_union : ∀ x ∈ K1 ⊔ K2, 1 ≤ (f1 + f2) x
⊢ rieszContentAux Λ K1 + (rieszContentAux Λ K2 + ε) = rieszContentAux Λ K1 + rieszContentAux Λ K2 + ε
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K₁ K₂ : Compacts X
disj : Disjoint ↑K₁ ↑K₂
b : ℝ≥0
f : X →C_c ℝ≥0
hf : f ∈ {f | ∀ x ∈ K₁ ⊔ K₂, 1 ≤ f x}
Λf_eq_b : Λ f = b
hsuppf : ∀ x ∈ K₁ ⊔ K₂, x ∈ support ⇑f
hsubsuppf : ↑K₁ ∪ ↑K₂ ⊆ tsupport ⇑f
g₁ g₂ : X →C_c ℝ≥0
hg₁ : EqOn (⇑g₁) 1 K₁.carrier
hg₂ : EqOn (⇑g₂) 1 K₂.carrier
sum_g : EqOn (⇑(g₁ + g₂)) 1 (tsupport f.toFun)
f_eq_sum : f = g₁ * f + g₂ * f
x : X
x_in_K₁ : x ∈ K₁
⊢ 1 ≤ (g₁ * f) x

Tactic:
simp [hg₁ x_in_K₁, hf x (mem_union_left _ x_in_K₁)]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: 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
hf : ∀ (x : X), 0 ≤ f x
K : Set X
hK : IsCompact K
hfK : ∀ x ∈ K, f x = 1
p : NNReal
hp : Λ f = ↑p
⊢ p ∈ ⇑(toNNRealLinear Λ) '' {f | ∀ x ∈ { carrier := K, isCompact' := hK }, 1 ≤ f x}

Tactic:
Set.mem_image

Goal state after:
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
hf : ∀ (x : X), 0 ≤ f x
K : Set X
hK : IsCompact K
hfK : ∀ x ∈ K, f x = 1
p : NNReal
hp : Λ f = ↑p
⊢ ∃ x ∈ {f | ∀ x ∈ { carrier := K, isCompact' := hK }, 1 ≤ f x}, (toNNRealLinear Λ) x = p
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.NNReal

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
f : X →C_c ℝ≥0
⊢ ∫⁻ (x : X), ↑(f x) ∂rieszMeasure Λ = ↑(Λ f)

Tactic:
by
  rw [lintegral_coe_eq_integral, ← ENNReal.ofNNReal_toNNReal]
  · rw [ENNReal.coe_inj, Real.toNNReal_of_nonneg (MeasureTheory.integral_nonneg (by intro a; simp)),
       NNReal.eq_iff, NNReal.coe_mk]
    exact integral_rieszMeasure Λ f
  rw [rieszMeasure]
  exact Continuous.integrable_of_hasCompactSupport (by fun_prop)
    (HasCompactSupport.comp_left f.hasCompactSupport rfl)

Goal state after:
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case h.refine_2
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 := a + ↑N * ε
y : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := fun n ↦ ⇑f ⁻¹' Ioc (y n - ε) (y n) ∩ tsupport ⇑f
m : Fin N
a✝¹ : m ∈ univ
n : Fin N
a✝ : n ∈ univ
⊢ m ≠ n → (Disjoint on E) m n

Tactic:
hmn

Goal state after:
case h.refine_2
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 := a + ↑N * ε
y : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := fun n ↦ ⇑f ⁻¹' Ioc (y n - ε) (y n) ∩ tsupport ⇑f
m : Fin N
a✝¹ : m ∈ univ
n : Fin N
a✝ : n ∈ univ
hmn : m ≠ n
⊢ (Disjoint on E) m n
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: 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
hε' : (b - a) / ↑N * (2 * μ.real K + |a| + b + (b - a) / ↑N) ≤ ε
⊢ Λ f ≤ ∫ (x : X), f x ∂rieszMeasure Λ + ε

Tactic:
let ε' := (b - a) / N

Goal state after:
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
hε' : (b - a) / ↑N * (2 * μ.real K + |a| + b + (b - a) / ↑N) ≤ ε
ε' : ℝ := (b - a) / ↑N
⊢ Λ f ≤ ∫ (x : X), f x ∂rieszMeasure Λ + ε
Module: Mathlib.MeasureTheory.Measure.FiniteMeasureExt

Goal state before:
pp: E : Type u_1
𝕜 : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : MeasurableSpace E
inst✝⁵ : PseudoEMetricSpace E
inst✝⁴ : BorelSpace E
inst✝³ : CompleteSpace E
inst✝² : SecondCountableTopology E
P P' : Measure E
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
A : StarSubalgebra 𝕜 (E →ᵇ 𝕜)
hA : (StarSubalgebra.map (toContinuousMapStarₐ 𝕜) A).SeparatesPoints
heq : ∀ g ∈ A, ∫ (x : E), g x ∂P = ∫ (x : E), g x ∂P'
A_toReal : Subalgebra ℝ (E →ᵇ ℝ) :=
  Subalgebra.comap (AlgHom.compLeftContinuousBounded ℝ ofRealAm ⋯) (Subalgebra.restrictScalars ℝ A.toSubalgebra)
hA_toReal : (Subalgebra.map (toContinuousMapₐ ℝ) A_toReal).SeparatesPoints
heq' : ∀ g ∈ A_toReal, ∫ (x : E), g x ∂P = ∫ (x : E), g x ∂P'
f : E →ᵇ ℝ
⊢ Tendsto (fun ε ↦ 6 * √ε) (𝓝[>] 0) (𝓝 (6 * 0))

Tactic:
apply tendsto_nhdsWithin_of_tendsto_nhds (Tendsto.const_mul 6 _)

Goal state after:
E : Type u_1
𝕜 : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : MeasurableSpace E
inst✝⁵ : PseudoEMetricSpace E
inst✝⁴ : BorelSpace E
inst✝³ : CompleteSpace E
inst✝² : SecondCountableTopology E
P P' : Measure E
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
A : StarSubalgebra 𝕜 (E →ᵇ 𝕜)
hA : (StarSubalgebra.map (toContinuousMapStarₐ 𝕜) A).SeparatesPoints
heq : ∀ g ∈ A, ∫ (x : E), g x ∂P = ∫ (x : E), g x ∂P'
A_toReal : Subalgebra ℝ (E →ᵇ ℝ) := ⋯
hA_toReal : (Subalgebra.map (toContinuousMapₐ ℝ) A_toReal).SeparatesPoints
heq' : ∀ g ∈ A_toReal, ∫ (x : E), g x ∂P = ∫ (x : E), g x ∂P'
f : E →ᵇ ℝ
⊢ Tendsto sqrt (𝓝 0) (𝓝 0)
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
hint : MemLp id 2 (Measure.map X P)
t : ℝ
this : IsProbabilityMeasure (Measure.map X P)
⊢ 1 + ↑(∫ (x : Ω), X x ∂P) * ↑t * I - ↑(∫ (x : Ω), X x ^ 2 ∂P) * ↑t ^ 2 / 2 =
    1 + ↑t * I * ↑(∫ (x : ℝ), x ∂Measure.map X P) + 2⁻¹ * (↑t * I) ^ 2 * ↑(∫ (x : ℝ), x ^ 2 ∂Measure.map X P)

Tactic:
rw [integral_map, integral_map]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
hint : MemLp id 2 (Measure.map X P)
t : ℝ
this : IsProbabilityMeasure (Measure.map X P)
⊢ 1 + ↑(∫ (x : Ω), X x ∂P) * ↑t * I - ↑(∫ (x : Ω), X x ^ 2 ∂P) * ↑t ^ 2 / 2 =
    1 + ↑t * I * ↑(∫ (x : Ω), X x ∂P) + 2⁻¹ * (↑t * I) ^ 2 * ↑(∫ (x : Ω), X x ^ 2 ∂P)

case hφ
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
hint : MemLp id 2 (Measure.map X P)
t : ℝ
this : IsProbabilityMeasure (Measure.map X P)
⊢ AEMeasurable X P

case hfm
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
hint : MemLp id 2 (Measure.map X P)
t : ℝ
this : IsProbabilityMeasure (Measure.map X P)
⊢ AEStronglyMeasurable (fun x ↦ x ^ 2) (Measure.map X P)

case hφ
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
hint : MemLp id 2 (Measure.map X P)
t : ℝ
this : IsProbabilityMeasure (Measure.map X P)
⊢ AEMeasurable X P

case hfm
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
hint : MemLp id 2 (Measure.map X P)
t : ℝ
this : IsProbabilityMeasure (Measure.map X P)
⊢ AEStronglyMeasurable (fun x ↦ x) (Measure.map X P)
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
μ : Measure X
inst✝ : IsFiniteMeasure μ
K : Set X
hK : IsCompact K
h : μ Kᶜ = 0
⊢ ∃ ν, ν.InnerRegular ∧ IsFiniteMeasure ν ∧ ν Kᶜ = 0 ∧ ∀ (g : X →ᵇ ℝ), ∫ (x : X), g x ∂μ = ∫ (x : X), g x ∂ν

Tactic:
by
  let μ' : Measure K := μ.comap Subtype.val
  obtain ⟨ν', ν'_reg, ν'_fin, hν'⟩ : ∃ (ν : Measure K), ν.Regular ∧ IsFiniteMeasure ν ∧
      ∀ g : K →ᵇ ℝ, ∫ x, g x ∂μ' = ∫ x, g x ∂ν := by
    have : CompactSpace K := isCompact_iff_compactSpace.mp hK
    exact Measure.exists_regular_eq_of_compactSpace μ'
  refine ⟨ν'.map Subtype.val, Measure.InnerRegular.map_of_continuous (by fun_prop),
    by infer_instance, ?_, fun g ↦ ?_⟩
  · rw [Measure.map_apply (by fun_prop) hK.measurableSet.compl]
    simp
  convert! hν' (g.compContinuous ⟨Subtype.val, by fun_prop⟩)
  · simp only [BoundedContinuousFunction.compContinuous_apply, ContinuousMap.coe_mk]
    rw [← integral_map (φ := Subtype.val) (by fun_prop) (by fun_prop)]
    simp only [map_comap_subtype_coe hK.measurableSet, μ', Measure.restrict_eq_self_of_ae_mem h]
  · rw [integral_map (φ := Subtype.val) (by fun_prop) (by fun_prop)]
    simp

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

Goal state before:
pp: μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
⊢ 2 * ∫ (x : ℝ) in {x | 2 < |2 * r⁻¹ * x|}, 1 - sinc (2 * r⁻¹ * x) ∂μ ≤ 2 * ∫ (x : ℝ), 1 - sinc (2 * r⁻¹ * x) ∂μ

Tactic:
grw [setIntegral_le_integral (by fun_prop) <| ae_of_all _ fun x ↦ ?_]

Goal state after:
μ : Measure ℝ
r : ℝ
inst✝ : IsProbabilityMeasure μ
hr : 0 < r
integrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ
x : ℝ
⊢ 0 x ≤ 1 - sinc (2 * r⁻¹ * x)
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝ : PseudoMetricSpace E
hS : IsTightMeasureSet S
x : E
⊢ Tendsto ((⨆ μ ∈ S, ⇑μ) ∘ fun r ↦ (Metric.closedBall x r)ᶜ) atTop (𝓝 0)

Tactic:
refine hS.comp <| .mono_right ?_ <| monotone_smallSets Metric.cobounded_le_cocompact

Goal state after:
E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝ : PseudoMetricSpace E
hS : IsTightMeasureSet S
x : E
⊢ Tendsto (fun r ↦ (Metric.closedBall x r)ᶜ) atTop (Bornology.cobounded E).smallSets
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝¹ : PseudoMetricSpace E
inst✝ : ProperSpace E
x : E
h✝ : ∀ ε > 0, ∃ N, ∀ n ≥ N, ⨆ μ ∈ S, μ (Metric.closedBall x n)ᶜ ≤ ε
ε : ℝ≥0∞
hε : 0 < ε
r : ℝ
h : ∀ n ≥ r, ⨆ μ ∈ S, μ (Metric.closedBall x n)ᶜ ≤ ε
⊢ ∃ K, IsCompact K ∧ ∀ μ ∈ S, μ Kᶜ ≤ ε

Tactic:
exact ⟨Metric.closedBall x r, isCompact_closedBall x r, by simpa using h r le_rfl⟩

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

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝³ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
h : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖x‖}) atTop (𝓝 0)
y : E
this : ProperSpace E
hy : y = 0
r : ℝ
hr : 0 ≤ r
⊢ ⨆ μ ∈ S, μ {x | r < ‖0‖} = 0

Tactic:
simp [not_lt.mpr hr]

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

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
⊢ IsTightMeasureSet (Set.range μ) ↔
    ∀ (y : E), Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)

Tactic:
by
  refine ⟨fun h z ↦ ?_, isTightMeasureSet_range_of_tendsto_limsup_inner 𝕜⟩
  rw [isTightMeasureSet_iff_inner_tendsto 𝕜] at h
  refine tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds (h z)
    (fun _ ↦ zero_le) fun r ↦ ?_
  simp_rw [iSup_range]
  exact limsup_le_iSup

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

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
y : E
hy : y = 0
r : ℝ
hr : 0 ≤ r
⊢ limsup (fun n ↦ (μ n) {x | r < ‖0‖}) atTop = 0

Tactic:
simp [not_lt.mpr hr]

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetSemiring C
h'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
h : mα = generateFrom C
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
hε : 0 < ε
⊢ generateFrom C = generateFrom (supClosure C)

Tactic:
apply le_antisymm (generateFrom_mono subset_supClosure)

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
hC : IsSetSemiring C
h'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0
h : mα = generateFrom C
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
hε : 0 < ε
⊢ generateFrom (supClosure C) ≤ generateFrom C
Module: Mathlib.MeasureTheory.Measure.PreVariation

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
s : Set X
hs : MeasurableSet s
h : preVariationFun f s ≠ ∞
ε : ℝ≥0
hε : 0 < ↑ε
h'ε : ↑ε < ∞
⊢ 0 < ε

Tactic:
simpa using hε

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsZeroOneMeasure μ
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
hμs : μ s = 1
hμt : μ t = 1
this : μ (s ∩ t) ≤ μ s
⊢ μ (s ∩ t) = 1

Tactic:
have : μ (s ∩ t) ≤ μ t := measure_mono inter_subset_right

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsZeroOneMeasure μ
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
hμs : μ s = 1
hμt : μ t = 1
this✝ : μ (s ∩ t) ≤ μ s
this : μ (s ∩ t) ≤ μ t
⊢ μ (s ∩ t) = 1
Module: Mathlib.MeasureTheory.Measure.Typeclasses.ZeroOne

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsZeroOneMeasure μ
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
this : μ (s ∩ t) ≤ μ s
⊢ μ (s ∩ t) = μ s * μ t

Tactic:
have : μ (s ∩ t) ≤ μ t := measure_mono inter_subset_right

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : IsZeroOneMeasure μ
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
this✝ : μ (s ∩ t) ≤ μ s
this : μ (s ∩ t) ≤ μ t
⊢ μ (s ∩ t) = μ s * μ t
Module: Mathlib.MeasureTheory.Order.UpperLower

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
hs : IsUpperSet s
x : ι → ℝ
hx : x ∈ frontier s
⊢ x ∈
    {x |
        (fun x ↦
            Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0)
              (𝓝 ((closure s).indicator 1 x)))
          x}ᶜ

Tactic:
by_cases h : x ∈ closure s

Goal state after:
case pos
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
hs : IsUpperSet s
x : ι → ℝ
hx : x ∈ frontier s
h : x ∈ closure s
⊢ x ∈
    {x |
        (fun x ↦
            Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0)
              (𝓝 ((closure s).indicator 1 x)))
          x}ᶜ

case neg
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
hs : IsUpperSet s
x : ι → ℝ
hx : x ∈ frontier s
h : x ∉ closure s
⊢ x ∈
    {x |
        (fun x ↦
            Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0)
              (𝓝 ((closure s).indicator 1 x)))
          x}ᶜ