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

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
⊢ Tendsto (fun i ↦ ∫ (x : X) in s i, f x ∂μ) atTop (𝓝 (∫ (x : X) in S, f x ∂μ))

Tactic:
]

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
⊢ Tendsto (fun i ↦ ∫ (x : X) in s i, f x ∂μ) atTop (𝓝 (∫ (x : X) in S, f x ∂μ))
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inl
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
cv : Icc a b ⊆ ⋃ i ∈ ↑s, Iotop (c i) (d i)
ab : b ≤ a
⊢ 0 ≤ ∑ i ∈ s, ofReal (↑f (d i) - ↑f (c i))

Tactic:
]

Goal state after:
case inl
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
cv : Icc a b ⊆ ⋃ i ∈ ↑s, Iotop (c i) (d i)
ab : b ≤ a
⊢ 0 ≤ ∑ i ∈ s, ofReal (↑f (d i) - ↑f (c i))
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this✝ : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
ν : Measure X := μ.withDensity fun x ↦ ‖f x‖ₑ
hν : ν = μ.withDensity fun x ↦ ‖f x‖ₑ
ε : ℝ≥0
ε0 : 0 < ↑ε
this : ∀ᶠ (i : ι) in atTop, ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
i : ι
hi : ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
⊢ ↑‖∫ (x : X) in S \ s i, f x ∂μ‖₊ ≤ ↑ε

Tactic:
]

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this✝ : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
ν : Measure X := μ.withDensity fun x ↦ ‖f x‖ₑ
hν : ν = μ.withDensity fun x ↦ ‖f x‖ₑ
ε : ℝ≥0
ε0 : 0 < ↑ε
this : ∀ᶠ (i : ι) in atTop, ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
i : ι
hi : ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
⊢ ↑‖∫ (x : X) in S \ s i, f x ∂μ‖₊ ≤ ↑ε
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
ab : a ≤ b
i : ℕ
cv : Icc a b ⊆ ⋃ i_1 ∈ ↑(insert i (s.erase i)), Iotop (c i_1) (d i_1)
is : i ∈ s
bcd : b ∈ Iotop (c i) (d i)
⊢ ofReal (↑f b - ↑f a) ≤ ∑ i ∈ insert i (s.erase i), ofReal (↑f (d i) - ↑f (c i))

Tactic:
]

Goal state after:
case inr
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
ab : a ≤ b
i : ℕ
cv : Icc a b ⊆ ⋃ i_1 ∈ ↑(insert i (s.erase i)), Iotop (c i_1) (d i_1)
is : i ∈ s
bcd : b ∈ Iotop (c i) (d i)
⊢ ofReal (↑f b - ↑f a) ≤ ∑ i ∈ insert i (s.erase i), ofReal (↑f (d i) - ↑f (c i))
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
ab : a ≤ b
i : ℕ
cv : Icc a b ⊆ Iotop (c i) (d i) ∪ ⋃ x ∈ ↑(s.erase i), Iotop (c x) (d x)
is : i ∈ s
bcd : b ∈ Iotop (c i) (d i)
⊢ ofReal (↑f b - ↑f a) ≤ ∑ i ∈ insert i (s.erase i), ofReal (↑f (d i) - ↑f (c i))

Tactic:
]

Goal state after:
case inr
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
ab : a ≤ b
i : ℕ
cv : Icc a b ⊆ Iotop (c i) (d i) ∪ ⋃ x ∈ ↑(s.erase i), Iotop (c x) (d x)
is : i ∈ s
bcd : b ∈ Iotop (c i) (d i)
⊢ ofReal (↑f b - ↑f a) ≤ ∑ i ∈ insert i (s.erase i), ofReal (↑f (d i) - ↑f (c i))
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
ab : a ≤ b
i : ℕ
cv : Icc a b ⊆ Iotop (c i) (d i) ∪ ⋃ x ∈ ↑(s.erase i), Iotop (c x) (d x)
is : i ∈ s
bcd : b ∈ Iotop (c i) (d i)
⊢ ofReal (↑f b - ↑f a) ≤ ofReal (↑f (d i) - ↑f (c i)) + ∑ x ∈ s.erase i, ofReal (↑f (d x) - ↑f (c x))

Tactic:
]

Goal state after:
case inr
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
ab : a ≤ b
i : ℕ
cv : Icc a b ⊆ Iotop (c i) (d i) ∪ ⋃ x ∈ ↑(s.erase i), Iotop (c x) (d x)
is : i ∈ s
bcd : b ∈ Iotop (c i) (d i)
⊢ ofReal (↑f b - ↑f a) ≤ ofReal (↑f (d i) - ↑f (c i)) + ∑ x ∈ s.erase i, ofReal (↑f (d x) - ↑f (c x))
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
ab : a ≤ b
i : ℕ
cv : Icc a b ⊆ Iotop (c i) (d i) ∪ ⋃ x ∈ ↑(s.erase i), Iotop (c x) (d x)
is : i ∈ s
bcd : b ∈ Ioc (c i) (d i)
⊢ ofReal (↑f b - ↑f a) ≤ ofReal (↑f (d i) - ↑f (c i) + (↑f (c i) - ↑f a))

pp: case a
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
ab : a ≤ b
i : ℕ
cv : Icc a b ⊆ Iotop (c i) (d i) ∪ ⋃ x ∈ ↑(s.erase i), Iotop (c x) (d x)
is : i ∈ s
bcd : b ∈ Ioc (c i) (d i)
⊢ Icc a (c i) ⊆ ⋃ i_1 ∈ ↑(s.erase i), Iotop (c i_1) (d i_1)

Tactic:
]

Goal state after:
case inr
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
ab : a ≤ b
i : ℕ
cv : Icc a b ⊆ Iotop (c i) (d i) ∪ ⋃ x ∈ ↑(s.erase i), Iotop (c x) (d x)
is : i ∈ s
bcd : b ∈ Ioc (c i) (d i)
⊢ ofReal (↑f b - ↑f a) ≤ ofReal (↑f (d i) - ↑f (c i) + (↑f (c i) - ↑f a))

case a
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
ab : a ≤ b
i : ℕ
cv : Icc a b ⊆ Iotop (c i) (d i) ∪ ⋃ x ∈ ↑(s.erase i), Iotop (c x) (d x)
is : i ∈ s
bcd : b ∈ Ioc (c i) (d i)
⊢ Icc a (c i) ⊆ ⋃ i_1 ∈ ↑(s.erase i), Iotop (c i_1) (d i_1)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this✝ : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
ν : Measure X := μ.withDensity fun x ↦ ‖f x‖ₑ
hν : ν = μ.withDensity fun x ↦ ‖f x‖ₑ
ε : ℝ≥0
ε0 : 0 < ↑ε
this : ∀ᶠ (i : ι) in atTop, ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
i : ι
hi : ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
hsm' : NullMeasurableSet (s i) ν
⊢ ν S - ν (s i) ≤ ↑ε

pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this✝ : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
ν : Measure X := μ.withDensity fun x ↦ ‖f x‖ₑ
hν : ν = μ.withDensity fun x ↦ ‖f x‖ₑ
ε : ℝ≥0
ε0 : 0 < ↑ε
this : ∀ᶠ (i : ι) in atTop, ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
i : ι
hi : ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
hsm' : NullMeasurableSet (s i) ν
⊢ ν (s i) ≠ ∞

Tactic:
]

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this✝ : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
ν : Measure X := μ.withDensity fun x ↦ ‖f x‖ₑ
hν : ν = μ.withDensity fun x ↦ ‖f x‖ₑ
ε : ℝ≥0
ε0 : 0 < ↑ε
this : ∀ᶠ (i : ι) in atTop, ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
i : ι
hi : ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
hsm' : NullMeasurableSet (s i) ν
⊢ ν S - ν (s i) ≤ ↑ε

X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this✝ : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
ν : Measure X := μ.withDensity fun x ↦ ‖f x‖ₑ
hν : ν = μ.withDensity fun x ↦ ‖f x‖ₑ
ε : ℝ≥0
ε0 : 0 < ↑ε
this : ∀ᶠ (i : ι) in atTop, ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
i : ι
hi : ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
hsm' : NullMeasurableSet (s i) ν
⊢ ν (s i) ≠ ∞
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
ab : a ≤ b
i : ℕ
cv : Icc a b ⊆ Iotop (c i) (d i) ∪ ⋃ x ∈ ↑(s.erase i), Iotop (c x) (d x)
is : i ∈ s
bcd : b ∈ Ioc (c i) (d i)
⊢ ofReal (↑f b - ↑f a) ≤ ofReal (↑f (d i) - ↑f a)

Tactic:
]

Goal state after:
case inr
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
ab : a ≤ b
i : ℕ
cv : Icc a b ⊆ Iotop (c i) (d i) ∪ ⋃ x ∈ ↑(s.erase i), Iotop (c x) (d x)
is : i ∈ s
bcd : b ∈ Ioc (c i) (d i)
⊢ ofReal (↑f b - ↑f a) ≤ ofReal (↑f (d i) - ↑f a)
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
ab : a ≤ b
i : ℕ
cv : Icc a b ⊆ Iotop (c i) (d i) ∪ ⋃ x ∈ ↑(s.erase i), Iotop (c x) (d x)
is : i ∈ s
bcd : b ∈ Ioc (c i) (d i)
⊢ ofReal (↑f (d i) - ↑f a) ≤ ofReal (↑f (d i) - ↑f a)

Tactic:
]

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

Goal state before:
pp: case refine_1
R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
⊢ f.outer (Ioc a b) ≤ f.length (Ioc a b)

Tactic:
]

Goal state after:
case refine_1
R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
⊢ f.outer (Ioc a b) ≤ f.length (Ioc a b)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case e'_5
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), MeasurableSet (s i)
h_anti : Antitone s
hne : atTop.NeBot
this✝ : IsDirectedOrder ι
i₀ : ι
hi₀ : IntegrableOn f (s i₀) μ
this :
  Tendsto (fun x ↦ ∫ (x : X) in s i₀, f x ∂μ - ∫ (x : X) in s i₀ \ s x, f x ∂μ) atTop
    (𝓝 (∫ (x : X) in s i₀, f x ∂μ - ∫ (x : X) in ⋃ i, s i₀ \ s i, f x ∂μ))
⊢ ∫ (x : X) in ⋂ n, s n, f x ∂μ = ∫ (x : X) in ⋂ i, s i, f x ∂μ

pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), MeasurableSet (s i)
h_anti : Antitone s
hne : atTop.NeBot
this✝ : IsDirectedOrder ι
i₀ : ι
hi₀ : IntegrableOn f (s i₀) μ
this :
  Tendsto (fun x ↦ ∫ (x : X) in s i₀, f x ∂μ - ∫ (x : X) in s i₀ \ s x, f x ∂μ) atTop
    (𝓝 (∫ (x : X) in s i₀, f x ∂μ - ∫ (x : X) in ⋃ i, s i₀ \ s i, f x ∂μ))
⊢ MeasurableSet (⋂ i, s i)

Tactic:
]

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), MeasurableSet (s i)
h_anti : Antitone s
hne : atTop.NeBot
this✝ : IsDirectedOrder ι
i₀ : ι
hi₀ : IntegrableOn f (s i₀) μ
this :
  Tendsto (fun x ↦ ∫ (x : X) in s i₀, f x ∂μ - ∫ (x : X) in s i₀ \ s x, f x ∂μ) atTop
    (𝓝 (∫ (x : X) in s i₀, f x ∂μ - ∫ (x : X) in ⋃ i, s i₀ \ s i, f x ∂μ))
⊢ MeasurableSet (⋂ i, s i)
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
A : ContinuousWithinAt (fun r ↦ ↑f r - ↑f a) (Ioi a) a
⊢ 0 < ↑δ

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
A : ContinuousWithinAt (fun r ↦ ↑f r - ↑f a) (Ioi a) a
⊢ 0 < ↑δ
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case convert_2
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), MeasurableSet (s i)
h_anti : Antitone s
hne : atTop.NeBot
this✝ : IsDirectedOrder ι
i₀ : ι
hi₀ : IntegrableOn f (s i₀) μ
this :
  Tendsto (fun x ↦ ∫ (x : X) in s i₀, f x ∂μ - ∫ (x : X) in s i₀ \ s x, f x ∂μ) atTop
    (𝓝 (∫ (x : X) in s i₀, f x ∂μ - ∫ (x : X) in ⋃ i, s i₀ \ s i, f x ∂μ))
i : ι
hi : i₀ ≤ i
⊢ ∫ (x : X) in s i, f x ∂μ = ∫ (x : X) in s i, f x ∂μ

Tactic:
]

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

Goal state before:
pp: G : Type w
inst✝ : TopologicalSpace G
μ : Content G
⊢ μ.innerContent ⊥ ≤ μ ⊥

Tactic:
]

Goal state after:
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
⊢ μ.innerContent ⊥ ≤ μ ⊥
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝ : TopologicalSpace G
μ : Content G
K : Compacts G
hK : ↑K ⊆ ↑⊥
⊢ ∅ = ∅

Tactic:
]

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

Goal state before:
pp: G : Type w
inst✝ : TopologicalSpace G
μ : Content G
K : Compacts G
hK : ↑K ⊆ ↑⊥
this : K = ⊥
⊢ μ ⊥ ≤ μ ⊥

Tactic:
]

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

Goal state before:
pp: G : Type w
inst✝ : TopologicalSpace G
μ : Content G
⊢ μ.innerContent ⊥ = 0

Tactic:
rw [← nonpos_iff_eq_zero, ← μ.empty]
  refine iSup₂_le fun K hK => ?_
  have : K = ⊥ := by
    ext1
    rw [subset_empty_iff.mp hK, Compacts.coe_bot]
  rw [this]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case refine_2
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), MeasurableSet (s i)
h_anti : Antitone s
hne : atTop.NeBot
this : IsDirectedOrder ι
i₀ : ι
hi₀ : IntegrableOn f (s i₀) μ
⊢ IntegrableOn f (s i₀ \ ⋂ i, s i) μ

Tactic:
]

Goal state after:
case refine_2
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), MeasurableSet (s i)
h_anti : Antitone s
hne : atTop.NeBot
this : IsDirectedOrder ι
i₀ : ι
hi₀ : IntegrableOn f (s i₀) μ
⊢ IntegrableOn f (s i₀ \ ⋂ i, s i) μ
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: case inr
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U✝ : Opens G
hU : μ.innerContent U✝ ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : ↑ε ≠ 0
h : ↑ε < μ.innerContent U✝
U : Compacts G
h1U : ↑U ⊆ ↑U✝
h2U : μ.innerContent U✝ - ↑ε < μ U
⊢ μ.innerContent U✝ - ↑ε ≤ μ U

Tactic:
]

Goal state after:
case inr
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
U✝ : Opens G
hU : μ.innerContent U✝ ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
h'ε : ↑ε ≠ 0
h : ↑ε < μ.innerContent U✝
U : Compacts G
h1U : ↑U ⊆ ↑U✝
h2U : μ.innerContent U✝ - ↑ε < μ U
⊢ μ.innerContent U✝ - ↑ε ≤ μ U
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case neg
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ, x ∈ t → f x = 0
hf : ¬AEStronglyMeasurable f (μ.restrict t)
⊢ 0 = 0

pp: case neg
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ, x ∈ t → f x = 0
hf : ¬AEStronglyMeasurable f (μ.restrict t)
⊢ ¬Integrable f (μ.restrict t)

Tactic:
]

Goal state after:
case neg
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ, x ∈ t → f x = 0
hf : ¬AEStronglyMeasurable f (μ.restrict t)
⊢ ¬Integrable f (μ.restrict t)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ, x ∈ t → f x = 0
hf : AEStronglyMeasurable f (μ.restrict t)
⊢ ∀ᵐ (x : X) ∂μ, x ∈ t → AEStronglyMeasurable.mk f hf x = 0 x

Tactic:
]

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ, x ∈ t → f x = 0
hf : AEStronglyMeasurable f (μ.restrict t)
⊢ ∀ᵐ (x : X) ∂μ, x ∈ t → AEStronglyMeasurable.mk f hf x = 0 x
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: case refine_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
t : Finset ℕ
K : ℕ → Compacts G
n : ℕ
s : Finset ℕ
hn : n ∉ s
ih : μ (s.sup K) ≤ ∑ i ∈ s, μ (K i)
⊢ μ (K n) + ∑ i ∈ s, μ (K i) ≤ μ (K n) + ∑ x ∈ s, μ (K x)

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ, x ∈ t → f x = 0
hf : AEStronglyMeasurable f (μ.restrict t)
x : X
hx : x ∈ t → f x = AEStronglyMeasurable.mk f hf x
h'x : x ∈ t → f x = 0
h''x : x ∈ t
⊢ f x = 0 x

Tactic:
]

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ, x ∈ t → f x = 0
hf : AEStronglyMeasurable f (μ.restrict t)
x : X
hx : x ∈ t → f x = AEStronglyMeasurable.mk f hf x
h'x : x ∈ t → f x = 0
h''x : x ∈ t
⊢ f x = 0 x
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
⊢ ↑K ⊆ ↑(⨆ i, U i)

Tactic:
]

Goal state after:
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
⊢ ↑K ⊆ ↑(⨆ i, U i)