mathlib-initiative/mathlib-tactics huggingface dataset:
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)
⊢ MeasurableSet (e (n + 1))

Tactic:
·

Goal state after:
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)
⊢ MeasurableSet (e (n + 1))
Module: Mathlib.MeasureTheory.Function.AEEqOfLIntegral

Goal state before:
pp: case refine_1
α : Type u_1
m0 : MeasurableSpace α
β : Type u_2
mβ : MeasurableSpace β
μ : Measure (α × β)
f g : α × β → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hf_int : ∫⁻ (x : α × β), f x ∂μ ≠ ∞
h :
  ∀ ⦃s : Set α⦄,
    MeasurableSet s →
      ∀ ⦃t : Set β⦄, MeasurableSet t → ∫⁻ (x : α × β) in s ×ˢ t, f x ∂μ = ∫⁻ (x : α × β) in s ×ˢ t, g x ∂μ
hg_int : ∫⁻ (x : α × β), g x ∂μ ≠ ∞
s : Set (α × β)
hs : MeasurableSet s
x✝ : μ s < ∞
⊢ Set.univ ∈ Set.image2 (fun x1 x2 ↦ x1 ×ˢ x2) {s | MeasurableSet s} {t | MeasurableSet t}

pp: case refine_2
α : Type u_1
m0 : MeasurableSpace α
β : Type u_2
mβ : MeasurableSpace β
μ : Measure (α × β)
f g : α × β → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hf_int : ∫⁻ (x : α × β), f x ∂μ ≠ ∞
h :
  ∀ ⦃s : Set α⦄,
    MeasurableSet s →
      ∀ ⦃t : Set β⦄, MeasurableSet t → ∫⁻ (x : α × β) in s ×ˢ t, f x ∂μ = ∫⁻ (x : α × β) in s ×ˢ t, g x ∂μ
hg_int : ∫⁻ (x : α × β), g x ∂μ ≠ ∞
s : Set (α × β)
hs : MeasurableSet s
x✝ : μ s < ∞
⊢ ∀ t ∈ Set.image2 (fun x1 x2 ↦ x1 ×ˢ x2) {s | MeasurableSet s} {t | MeasurableSet t},
    ∫⁻ (x : α × β) in t, f x ∂μ = ∫⁻ (x : α × β) in t, g x ∂μ

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
m0 : MeasurableSpace α
β : Type u_2
mβ : MeasurableSpace β
μ : Measure (α × β)
f g : α × β → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hf_int : ∫⁻ (x : α × β), f x ∂μ ≠ ∞
h :
  ∀ ⦃s : Set α⦄,
    MeasurableSet s →
      ∀ ⦃t : Set β⦄, MeasurableSet t → ∫⁻ (x : α × β) in s ×ˢ t, f x ∂μ = ∫⁻ (x : α × β) in s ×ˢ t, g x ∂μ
hg_int : ∫⁻ (x : α × β), g x ∂μ ≠ ∞
s : Set (α × β)
hs : MeasurableSet s
x✝ : μ s < ∞
⊢ Set.univ ∈ Set.image2 (fun x1 x2 ↦ x1 ×ˢ x2) {s | MeasurableSet s} {t | MeasurableSet t}
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: case inr.refine_2
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)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : v.Nonempty
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
⊢ ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x (8 * R0)

Tactic:
·

Goal state after:
case inr.refine_2
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)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : v.Nonempty
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
⊢ ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x (8 * R0)
Module: Mathlib.MeasureTheory.Function.AEEqOfLIntegral

Goal state before:
pp: case refine_2
α : Type u_1
m0 : MeasurableSpace α
β : Type u_2
mβ : MeasurableSpace β
μ : Measure (α × β)
f g : α × β → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hf_int : ∫⁻ (x : α × β), f x ∂μ ≠ ∞
h :
  ∀ ⦃s : Set α⦄,
    MeasurableSet s →
      ∀ ⦃t : Set β⦄, MeasurableSet t → ∫⁻ (x : α × β) in s ×ˢ t, f x ∂μ = ∫⁻ (x : α × β) in s ×ˢ t, g x ∂μ
hg_int : ∫⁻ (x : α × β), g x ∂μ ≠ ∞
s : Set (α × β)
hs : MeasurableSet s
x✝ : μ s < ∞
⊢ ∀ t ∈ Set.image2 (fun x1 x2 ↦ x1 ×ˢ x2) {s | MeasurableSet s} {t | MeasurableSet t},
    ∫⁻ (x : α × β) in t, f x ∂μ = ∫⁻ (x : α × β) in t, g x ∂μ

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
m0 : MeasurableSpace α
β : Type u_2
mβ : MeasurableSpace β
μ : Measure (α × β)
f g : α × β → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hf_int : ∫⁻ (x : α × β), f x ∂μ ≠ ∞
h :
  ∀ ⦃s : Set α⦄,
    MeasurableSet s →
      ∀ ⦃t : Set β⦄, MeasurableSet t → ∫⁻ (x : α × β) in s ×ˢ t, f x ∂μ = ∫⁻ (x : α × β) in s ×ˢ t, g x ∂μ
hg_int : ∫⁻ (x : α × β), g x ∂μ ≠ ∞
s : Set (α × β)
hs : MeasurableSet s
x✝ : μ s < ∞
⊢ ∀ t ∈ Set.image2 (fun x1 x2 ↦ x1 ×ˢ x2) {s | MeasurableSet s} {t | MeasurableSet t},
    ∫⁻ (x : α × β) in t, f x ∂μ = ∫⁻ (x : α × β) in t, g x ∂μ
Module: Mathlib.MeasureTheory.Function.AEEqOfLIntegral

Goal state before:
pp: case refine_1
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ∞
hfg : μ.withDensity f = μ.withDensity g
⊢ ∫⁻ (x : α), g x ∂μ ≠ ∞

pp: case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ∞
hfg : μ.withDensity f = μ.withDensity g
s : Set α
hs : MeasurableSet s
x✝ : μ s < ∞
⊢ ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ∞
hfg : μ.withDensity f = μ.withDensity g
⊢ ∫⁻ (x : α), g x ∂μ ≠ ∞
Module: Mathlib.MeasureTheory.Function.AEEqOfLIntegral

Goal state before:
pp: case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ∞
hfg : μ.withDensity f = μ.withDensity g
s : Set α
hs : MeasurableSet s
x✝ : μ s < ∞
⊢ ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ∞
hfg : μ.withDensity f = μ.withDensity g
s : Set α
hs : MeasurableSet s
x✝ : μ s < ∞
⊢ ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: case a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
⊢ sInf {d | μ ≤ d + 0} ≤ μ

pp: case a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
⊢ μ ≤ sInf {d | μ ≤ d + 0}

Tactic:
·

Goal state after:
case a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
⊢ sInf {d | μ ≤ d + 0} ≤ μ
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: case a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
⊢ μ ≤ sInf {d | μ ≤ d + 0}

Tactic:
·

Goal state after:
case a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
⊢ μ ≤ sInf {d | μ ≤ d + 0}
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: case a
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
inst✝ : IsFiniteMeasure ν
h₁ : MeasurableSet s
h₂ : ν ≤ μ
measure_sub : Measure α := ⋯
h_measure_sub_add : ν + measure_sub = μ
⊢ sInf {d | μ ≤ d + ν} ≤ measure_sub

pp: case a
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
inst✝ : IsFiniteMeasure ν
h₁ : MeasurableSet s
h₂ : ν ≤ μ
measure_sub : Measure α := ⋯
h_measure_sub_add : ν + measure_sub = μ
⊢ measure_sub ≤ sInf {d | μ ≤ d + ν}

Tactic:
·

Goal state after:
case a
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
inst✝ : IsFiniteMeasure ν
h₁ : MeasurableSet s
h₂ : ν ≤ μ
measure_sub : Measure α := ⋯
h_measure_sub_add : ν + measure_sub = μ
⊢ sInf {d | μ ≤ d + ν} ≤ measure_sub
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hμ : μ s ≠ 0
hs : NullMeasurableSet s μ
hμ₁ : μ s ≠ ∞
g : α → ℝ≥0∞
hint : ∫⁻ (a : α) in s, g a ∂μ ≠ ∞
hg : Measurable g
hgf : g ≤ f
hfg : ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, g a ∂μ
hfg' : ⨍⁻ (a : α) in s, f a ∂μ = ⨍⁻ (a : α) in s, g a ∂μ
this : 0 < (μ.restrict s) ({a | ⨍ (a : α) in s, (g a).toReal ∂μ ≤ (g a).toReal} \ {x | g x = ∞})
x : α
hfx : ⨍⁻ (x : α) in s, g x ∂μ ≤ g x
hx : x ∉ {x | g x = ∞}
⊢ x ∈ {a | ⨍⁻ (a : α) in s, g a ∂μ ≤ f a}

pp: case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hμ : μ s ≠ 0
hs : NullMeasurableSet s μ
hμ₁ : μ s ≠ ∞
g : α → ℝ≥0∞
hint : ∫⁻ (a : α) in s, g a ∂μ ≠ ∞
hg : Measurable g
hgf : g ≤ f
hfg : ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, g a ∂μ
hfg' : ⨍⁻ (a : α) in s, f a ∂μ = ⨍⁻ (a : α) in s, g a ∂μ
this : 0 < (μ.restrict s) ({a | ⨍ (a : α) in s, (g a).toReal ∂μ ≤ (g a).toReal} \ {x | g x = ∞})
x : α
hfx : ⨍ (a : α) in s, (g a).toReal ∂μ ≤ (g x).toReal
hx : x ∉ {x | g x = ∞}
⊢ ∀ᵐ (x : α) ∂μ.restrict s, g x ≠ ∞

Tactic:
·

Goal state after:
case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hμ : μ s ≠ 0
hs : NullMeasurableSet s μ
hμ₁ : μ s ≠ ∞
g : α → ℝ≥0∞
hint : ∫⁻ (a : α) in s, g a ∂μ ≠ ∞
hg : Measurable g
hgf : g ≤ f
hfg : ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, g a ∂μ
hfg' : ⨍⁻ (a : α) in s, f a ∂μ = ⨍⁻ (a : α) in s, g a ∂μ
this : 0 < (μ.restrict s) ({a | ⨍ (a : α) in s, (g a).toReal ∂μ ≤ (g a).toReal} \ {x | g x = ∞})
x : α
hfx : ⨍⁻ (x : α) in s, g x ∂μ ≤ g x
hx : x ∉ {x | g x = ∞}
⊢ x ∈ {a | ⨍⁻ (a : α) in s, g a ∂μ ≤ f a}
Module: Mathlib.MeasureTheory.Integral.Average

Goal state before:
pp: case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hμ : μ s ≠ 0
hs : NullMeasurableSet s μ
hμ₁ : μ s ≠ ∞
g : α → ℝ≥0∞
hint : ∫⁻ (a : α) in s, g a ∂μ ≠ ∞
hg : Measurable g
hgf : g ≤ f
hfg : ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, g a ∂μ
hfg' : ⨍⁻ (a : α) in s, f a ∂μ = ⨍⁻ (a : α) in s, g a ∂μ
this : 0 < (μ.restrict s) ({a | ⨍ (a : α) in s, (g a).toReal ∂μ ≤ (g a).toReal} \ {x | g x = ∞})
x : α
hfx : ⨍ (a : α) in s, (g a).toReal ∂μ ≤ (g x).toReal
hx : x ∉ {x | g x = ∞}
⊢ ∀ᵐ (x : α) ∂μ.restrict s, g x ≠ ∞

Tactic:
·

Goal state after:
case inr
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hμ : μ s ≠ 0
hs : NullMeasurableSet s μ
hμ₁ : μ s ≠ ∞
g : α → ℝ≥0∞
hint : ∫⁻ (a : α) in s, g a ∂μ ≠ ∞
hg : Measurable g
hgf : g ≤ f
hfg : ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in s, g a ∂μ
hfg' : ⨍⁻ (a : α) in s, f a ∂μ = ⨍⁻ (a : α) in s, g a ∂μ
this : 0 < (μ.restrict s) ({a | ⨍ (a : α) in s, (g a).toReal ∂μ ≤ (g a).toReal} \ {x | g x = ∞})
x : α
hfx : ⨍ (a : α) in s, (g a).toReal ∂μ ≤ (g x).toReal
hx : x ∉ {x | g x = ∞}
⊢ ∀ᵐ (x : α) ∂μ.restrict s, g x ≠ ∞
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: case a
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
⊢ sInf ((fun μ ↦ μ.restrict s) '' {d | μ ≤ d + ν}) ≤ sInf {d | μ.restrict s ≤ d + ν.restrict s}

pp: case a
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
⊢ sInf {d | μ.restrict s ≤ d + ν.restrict s} ≤ sInf ((fun μ ↦ μ.restrict s) '' {d | μ ≤ d + ν})

Tactic:
·

Goal state after:
case a
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
⊢ sInf ((fun μ ↦ μ.restrict s) '' {d | μ ≤ d + ν}) ≤ sInf {d | μ.restrict s ≤ d + ν.restrict s}
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: case a.refine_1
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
ν' : Measure α
h_ν'_in : μ.restrict s ≤ ν' + ν.restrict s
⊢ ν' + ⊤.restrict sᶜ ∈ {d | μ ≤ d + ν}

pp: case a.refine_2
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
ν' : Measure α
h_ν'_in : μ.restrict s ≤ ν' + ν.restrict s
⊢ (fun μ ↦ μ.restrict s) (ν' + ⊤.restrict sᶜ) = ν'.restrict s

Tactic:
·

Goal state after:
case a.refine_1
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
ν' : Measure α
h_ν'_in : μ.restrict s ≤ ν' + ν.restrict s
⊢ ν' + ⊤.restrict sᶜ ∈ {d | μ ≤ d + ν}
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: case a.refine_1.refine_1
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
ν' : Measure α
h_ν'_in : μ.restrict s ≤ ν' + ν.restrict s
t : Set α
h_meas_t : MeasurableSet t
⊢ μ (t ∩ s) ≤ (ν' + ν + ⊤.restrict sᶜ) (t ∩ s)

pp: case a.refine_1.refine_2
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
ν' : Measure α
h_ν'_in : μ.restrict s ≤ ν' + ν.restrict s
t : Set α
h_meas_t : MeasurableSet t
⊢ μ (t \ s) ≤ (ν' + ν + ⊤.restrict sᶜ) (t \ s)

Tactic:
·

Goal state after:
case a.refine_1.refine_1
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
ν' : Measure α
h_ν'_in : μ.restrict s ≤ ν' + ν.restrict s
t : Set α
h_meas_t : MeasurableSet t
⊢ μ (t ∩ s) ≤ (ν' + ν + ⊤.restrict sᶜ) (t ∩ s)
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: case a.refine_1.refine_2
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
ν' : Measure α
h_ν'_in : μ.restrict s ≤ ν' + ν.restrict s
t : Set α
h_meas_t : MeasurableSet t
⊢ μ (t \ s) ≤ (ν' + ν + ⊤.restrict sᶜ) (t \ s)

Tactic:
·

Goal state after:
case a.refine_1.refine_2
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
ν' : Measure α
h_ν'_in : μ.restrict s ≤ ν' + ν.restrict s
t : Set α
h_meas_t : MeasurableSet t
⊢ μ (t \ s) ≤ (ν' + ν + ⊤.restrict sᶜ) (t \ s)
Module: Mathlib.MeasureTheory.Measure.Decomposition.Hahn

Goal state before:
pp: case refine_2
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)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ m, ⋂ n, f m n
hγ : Tendsto (fun m ↦ γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)
hd : Tendsto (fun m ↦ d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n)))
m : ℕ
⊢ ∀ (n m_1 : ℕ), n ≤ m_1 → f m m_1 ⊆ f m n

Tactic:
·

Goal state after:
case refine_2
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)
le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ m, ⋂ n, f m n
hγ : Tendsto (fun m ↦ γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)
hd : Tendsto (fun m ↦ d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n)))
m : ℕ
⊢ ∀ (n m_1 : ℕ), n ≤ m_1 → f m m_1 ⊆ f m n
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: case a.refine_2
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
ν' : Measure α
h_ν'_in : μ.restrict s ≤ ν' + ν.restrict s
⊢ (fun μ ↦ μ.restrict s) (ν' + ⊤.restrict sᶜ) = ν'.restrict s

Tactic:
·

Goal state after:
case a.refine_2
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
ν' : Measure α
h_ν'_in : μ.restrict s ≤ ν' + ν.restrict s
⊢ (fun μ ↦ μ.restrict s) (ν' + ⊤.restrict sᶜ) = ν'.restrict s
Module: Mathlib.MeasureTheory.Measure.Sub

Goal state before:
pp: case a.refine_2
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
ν' : Measure α
h_ν'_in : μ.restrict s ≤ ν' + ν.restrict s
⊢ (fun μ ↦ μ.restrict s) (ν' + ⊤.restrict sᶜ) = ν'.restrict s

Tactic:
ext1 t h_meas_t
      simp [restrict_apply h_meas_t, restrict_apply (h_meas_t.inter h_meas_s), inter_assoc]

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

Goal state before:
pp: case a
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
⊢ sInf {d | μ.restrict s ≤ d + ν.restrict s} ≤ sInf ((fun μ ↦ μ.restrict s) '' {d | μ ≤ d + ν})

Tactic:
·

Goal state after:
case a
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
s : Set α
h_meas_s : MeasurableSet s
h_nonempty : {d | μ ≤ d + ν}.Nonempty
⊢ sInf {d | μ.restrict s ≤ d + ν.restrict s} ≤ sInf ((fun μ ↦ μ.restrict s) '' {d | μ ≤ d + ν})
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
h : μ.HaveLebesgueDecomposition ν
⊢ Measurable (μ.rnDeriv ν)

pp: case neg
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
h : ¬μ.HaveLebesgueDecomposition ν
⊢ Measurable (μ.rnDeriv ν)

Tactic:
·

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
h : μ.HaveLebesgueDecomposition ν
⊢ Measurable (μ.rnDeriv ν)
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
h : ¬μ.HaveLebesgueDecomposition ν
⊢ Measurable (μ.rnDeriv ν)

Tactic:
·

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
h : ¬μ.HaveLebesgueDecomposition ν
⊢ Measurable (μ.rnDeriv ν)
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case pos
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
h : μ.HaveLebesgueDecomposition ν
⊢ μ.singularPart ν ⟂ₘ ν

pp: case neg
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
h : ¬μ.HaveLebesgueDecomposition ν
⊢ μ.singularPart ν ⟂ₘ ν

Tactic:
·

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
h : μ.HaveLebesgueDecomposition ν
⊢ μ.singularPart ν ⟂ₘ ν
Module: Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
h : ¬μ.HaveLebesgueDecomposition ν
⊢ μ.singularPart ν ⟂ₘ ν

Tactic:
·

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
h : ¬μ.HaveLebesgueDecomposition ν
⊢ μ.singularPart ν ⟂ₘ ν