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

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure α
f : α → E
s : Set α
⊢ ⨍ (x : α) in s, f x ∂μ = (μ.real s)⁻¹ • ∫ (x : α) in s, f x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure α
f : α → E
s : Set α
⊢ ⨍ (x : α) in s, f x ∂μ = ∫ (x : α), f x ∂(μ s)⁻¹ • μ.restrict s

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure α
f g : α → E
h : f =ᵐ[μ] g
⊢ ⨍ (x : α), f x ∂μ = ⨍ (x : α), g x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure α
s t : Set α
f : α → E
h : s =ᵐ[μ] t
⊢ ⨍ (x : α) in s, f x ∂μ = ⨍ (x : α) in t, f x ∂μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
h : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ
⊢ f =ᵐ[μ] g

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure α
s : Set α
f g : α → E
hs : MeasurableSet s
h : ∀ᵐ (x : α) ∂μ, x ∈ s → f x = g x
⊢ ⨍ (x : α) in s, f x ∂μ = ⨍ (x : α) in s, g x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.Hahn

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d✝ : Set α → ℝ := fun s ↦ ↑(μ s).toNNReal - ↑(ν s).toNNReal
c✝ : Set ℝ := d✝ '' {s | MeasurableSet s}
γ : ℝ := sSup c✝
hμ : ∀ (s : Set α), μ s ≠ ∞
hν : ∀ (s : Set α), ν s ≠ ∞
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d✝ s = d✝ (s \ t) + d✝ (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n ↦ d✝ (s n)) atTop (𝓝 (d✝ (⋃ n, s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n ↦ d✝ (s n)) atTop (𝓝 (d✝ (⋂ n, s n)))
bdd_c : BddAbove c✝
c_nonempty : c✝.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d✝ s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d✝ s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d✝ (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d✝ (e n)
f : ℕ → ℕ → Set α := fun n m ↦ (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
a b c d : ℕ
hab : a ≤ b
hcd : c ≤ d
⊢ f a d ⊆ f b c

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
ν : Measure α
inst✝ : IsFiniteMeasure ν
f : α → E
hμ : Integrable f μ
hν : Integrable f ν
⊢ ⨍ (x : α), f x ∂(μ + ν) =
    (μ.real univ / (μ.real univ + ν.real univ)) • ⨍ (x : α), f x ∂μ +
      (ν.real univ / (μ.real univ + ν.real univ)) • ⨍ (x : α), f x ∂ν

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ∞
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
⊢ ∀ a ∈ t', r a ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure α
f : α → E
s : Set α
h : μ s ≠ ∞
⊢ μ.real s • ⨍ (x : α) in s, f x ∂μ = ∫ (x : α) in s, f x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure α
f : α → E
s t : Set α
hd : AEDisjoint μ s t
ht : NullMeasurableSet t μ
hsμ : μ s ≠ ∞
htμ : μ t ≠ ∞
hfs : IntegrableOn f s μ
hft : IntegrableOn f t μ
⊢ ⨍ (x : α) in s ∪ t, f x ∂μ =
    (μ.real s / (μ.real s + μ.real t)) • ⨍ (x : α) in s, f x ∂μ +
      (μ.real t / (μ.real s + μ.real t)) • ⨍ (x : α) in t, f x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure α
f : α → E
s t : Set α
hd : AEDisjoint μ s t
ht : NullMeasurableSet t μ
hs₀ : μ s ≠ 0
ht₀ : μ t ≠ 0
hsμ : μ s ≠ ∞
htμ : μ t ≠ ∞
hfs : IntegrableOn f s μ
hft : IntegrableOn f t μ
⊢ ⨍ (x : α) in s ∪ t, f x ∂μ ∈ openSegment ℝ (⨍ (x : α) in s, f x ∂μ) (⨍ (x : α) in t, f x ∂μ)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ∞
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
⊢ ∃ u ⊆ t', u.PairwiseDisjoint B ∧ ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.Hahn

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s ↦ ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ∞
hν : ∀ (s : Set α), ν s ≠ ∞
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n ↦ d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n ↦ d (s n)) atTop (𝓝 (d (⋂ n, s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m ↦ (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
n m : ℕ
hnm : n ≤ m
⊢ f n (m + 1) = f n m ∩ e (m + 1)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure α
f : α → E
s t : Set α
hd : AEDisjoint μ s t
ht : NullMeasurableSet t μ
hsμ : μ s ≠ ∞
htμ : μ t ≠ ∞
hfs : IntegrableOn f s μ
hft : IntegrableOn f t μ
⊢ ⨍ (x : α) in s ∪ t, f x ∂μ ∈ [⨍ (x : α) in s, f x ∂μ -[ℝ] ⨍ (x : α) in t, f x ∂μ]

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → E
s : Set α
hs : NullMeasurableSet s μ
hs₀ : μ s ≠ 0
hsc₀ : μ sᶜ ≠ 0
hfi : Integrable f μ
⊢ ⨍ (x : α), f x ∂μ ∈ openSegment ℝ (⨍ (x : α) in s, f x ∂μ) (⨍ (x : α) in sᶜ, f x ∂μ)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
h : NeZero μ
c : E
⊢ ⨍ (_x : α), c ∂μ = c

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → E
⊢ ∫ (x : α), ⨍ (a : α), f a ∂μ ∂μ = ∫ (x : α), f x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → E
⊢ ∫ (x : α), f x - ⨍ (a : α), f a ∂μ ∂μ = 0

Tactic:
by_cases hf : Integrable f μ
  · rw [integral_sub hf (integrable_const _), integral_average, sub_self]
  refine integral_undef fun h => hf ?_
  convert! h.add (integrable_const (⨍ a, f a ∂μ))
  exact (sub_add_cancel _ _).symm

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → E
⊢ ∫ (x : α), f x - ⨍ (a : α), f a ∂μ ∂μ = 0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
μ : Measure α
f : α → E
inst✝¹ : CompleteSpace E
inst✝ : IsFiniteMeasure μ
hf : Integrable f μ
⊢ ∫ (x : α), ⨍ (a : α), f a ∂μ - f x ∂μ = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ∞
hgi : ∫⁻ (x : α), g x ∂μ ≠ ∞
hfg : ∀ ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ
hf' : AEFinStronglyMeasurable f μ
hg' : AEFinStronglyMeasurable g μ
s : Set α := hf'.sigmaFiniteSet
t : Set α := hg'.sigmaFiniteSet
this : f =ᵐ[μ.restrict (s ∪ t)] g
⊢ f =ᵐ[μ] g

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
hf₀ : 0 ≤ᵐ[μ] f
⊢ ENNReal.ofReal (⨍ (x : α), f x ∂μ) = (∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ) / μ univ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ
hf : IntegrableOn f s μ
hf₀ : 0 ≤ᵐ[μ.restrict s] f
⊢ ENNReal.ofReal (⨍ (x : α) in s, f x ∂μ) = (∫⁻ (x : α) in s, ENNReal.ofReal (f x) ∂μ) / μ s

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ∞
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
x✝ : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}
vu : v ⊆ u
⊢ ∀ a ∈ v, dist (c a) x ≤ r a + R x

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
hf' : ∀ᵐ (x : α) ∂μ, f x ≠ ∞
⊢ (⨍⁻ (x : α), f x ∂μ).toReal = ⨍ (x : α), (f x).toReal ∂μ

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ∞
hgi : ∫⁻ (x : α), g x ∂μ ≠ ∞
hfg : ∀ ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ
⊢ f =ᵐ[μ] g

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hf : AEMeasurable f (μ.restrict s)
hf' : ∀ᵐ (x : α) ∂μ.restrict s, f x ≠ ∞
⊢ (⨍⁻ (x : α) in s, f x ∂μ).toReal = ⨍ (x : α) in s, (f x).toReal ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ∞
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
x✝ : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}
vu : v ⊆ u
Idist_v : ∀ a ∈ v, dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
⊢ BddAbove (r '' v)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ
hμ : μ s ≠ 0
hμ₁ : μ s ≠ ∞
hf : IntegrableOn f s μ
H : μ {x | x ∈ s ∧ f x ≤ ⨍ (a : α) in s, f a ∂μ} = 0
⊢ (μ.restrict s) {x | f x ≤ ⨍ (a : α) in s, f a ∂μ} = 0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.Hahn

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s ↦ ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ∞
hν : ∀ (s : Set α), ν s ≠ ∞
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n ↦ d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n ↦ d (s n)) atTop (𝓝 (d (⋂ n, s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m ↦ (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) = γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1)))

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.Hahn

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s ↦ ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ∞
hν : ∀ (s : Set α), ν s ≠ ∞
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n ↦ d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n ↦ d (s n)) atTop (𝓝 (d (⋂ n, s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m ↦ (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1))) =
    γ - (1 / 2) ^ (n + 1) + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ
hμ : μ s ≠ 0
hμ₁ : μ s ≠ ∞
hf : IntegrableOn f s μ
⊢ 0 < μ {x | x ∈ s ∧ f x ≤ ⨍ (a : α) in s, f a ∂μ}

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ
hμ : μ s ≠ 0
hμ₁ : μ s ≠ ∞
hf : IntegrableOn f s μ
⊢ 0 < μ {x | x ∈ s ∧ ⨍ (a : α) in s, f a ∂μ ≤ f x}

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
inst✝ : IsFiniteMeasure μ
hμ : μ ≠ 0
hf : Integrable f μ
⊢ 0 < μ {x | f x ≤ ⨍ (a : α), f a ∂μ}

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
inst✝ : IsFiniteMeasure μ
hμ : μ ≠ 0
hf : Integrable f μ
⊢ 0 < μ {x | ⨍ (a : α), f a ∂μ ≤ f x}

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.Hahn

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
d : Set α → ℝ := fun s ↦ ↑(μ s).toNNReal - ↑(ν s).toNNReal
c : Set ℝ := d '' {s | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), μ s ≠ ∞
hν : ∀ (s : Set α), ν s ≠ ∞
to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s
to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s
d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n ↦ d (s n)) atTop (𝓝 (d (⋃ n, s n)))
d_Inter :
  ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n ↦ d (s n)) atTop (𝓝 (d (⋂ n, s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun n m ↦ (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c
f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)
n✝ m : ℕ
h : m ≤ n✝
n : ℕ
hmn : m ≤ n
ih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
⊢ d (e (n + 1)) + d (f m n) ≤ d (e (n + 1)) + d (f m n \ e (n + 1)) + d (f m (n + 1))

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
N : Set α
f : α → ℝ
inst✝ : IsFiniteMeasure μ
hμ : μ ≠ 0
hf : Integrable f μ
hN : μ N = 0
⊢ ∃ x ∉ N, f x ≤ ⨍ (a : α), f a ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
N : Set α
f : α → ℝ
inst✝ : IsFiniteMeasure μ
hμ : μ ≠ 0
hf : Integrable f μ
hN : μ N = 0
⊢ ∃ x ∉ N, ⨍ (a : α), f a ∂μ ≤ f x

Tactic:
by

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set (Set α)
f g : α → ℝ≥0∞
h_eq✝ : m0 = MeasurableSpace.generateFrom s
h_inter : IsPiSystem s
basic : ∀ t ∈ s, ∫⁻ (x : α) in t, f x ∂μ = ∫⁻ (x : α) in t, g x ∂μ
h_univ : ∫⁻ (x : α), f x ∂μ = ∫⁻ (x : α), g x ∂μ
hf_int : ∫⁻ (x : α), f x ∂μ ≠ ∞
t : Set α
ht : MeasurableSet t
h_eq : ∫⁻ (x : α) in t, f x ∂μ = ∫⁻ (x : α) in t, g x ∂μ
⊢ ∫⁻ (x : α), g x ∂μ < ∞

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
inst✝ : IsProbabilityMeasure μ
hf : Integrable f μ
⊢ 0 < μ {x | f x ≤ ∫ (a : α), f a ∂μ}

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
inst✝ : IsProbabilityMeasure μ
hf : Integrable f μ
⊢ 0 < μ {x | ∫ (a : α), f a ∂μ ≤ f x}

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
inst✝ : IsProbabilityMeasure μ
hf : Integrable f μ
⊢ ∃ x, f x ≤ ∫ (a : α), f a ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
inst✝ : IsProbabilityMeasure μ
hf : Integrable f μ
⊢ ∃ x, ∫ (a : α), f a ∂μ ≤ f x

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
N : Set α
f : α → ℝ
inst✝ : IsProbabilityMeasure μ
hf : Integrable f μ
hN : μ N = 0
⊢ ∃ x ∉ N, f x ≤ ∫ (a : α), f a ∂μ

Tactic:
by

Goal state after: