l3lab/ntp-mathlib huggingface dataset:
theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b : X}
    (ha : IsLeast s a) (hb : IsGreatest s b) (hs : μ s ≠ ∞) (h's : MeasurableSet s) :
    IntegrableOn f s μ := by
  borelize E
  

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹¹ : MeasurableSpace X
inst✝¹⁰ : TopologicalSpace X
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
hs : ↑↑μ s ≠ ⊤
h's : MeasurableSet s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
⊢ IntegrableOn f s

Next tactic:
obtain rfl | _ := s.eq_empty_or_nonempty
theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b : X}
    (ha : IsLeast s a) (hb : IsGreatest s b) (hs : μ s ≠ ∞) (h's : MeasurableSet s) :
    IntegrableOn f s μ := by
  borelize E
  obtain rfl | _ := s.eq_empty_or_nonempty
  · 

Proof state:
case inl
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹¹ : MeasurableSpace X
inst✝¹⁰ : TopologicalSpace X
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
f g : X → E
μ : Measure X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
a b : X
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
hmono : MonotoneOn f ∅
ha : IsLeast ∅ a
hb : IsGreatest ∅ b
hs : ↑↑μ ∅ ≠ ⊤
h's : MeasurableSet ∅
⊢ IntegrableOn f ∅

Next tactic:
exact integrableOn_empty
theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b : X}
    (ha : IsLeast s a) (hb : IsGreatest s b) (hs : μ s ≠ ∞) (h's : MeasurableSet s) :
    IntegrableOn f s μ := by
  borelize E
  obtain rfl | _ := s.eq_empty_or_nonempty
  · exact integrableOn_empty
  

Proof state:
case inr
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹¹ : MeasurableSpace X
inst✝¹⁰ : TopologicalSpace X
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
hs : ↑↑μ s ≠ ⊤
h's : MeasurableSet s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
h✝ : Set.Nonempty s
⊢ IntegrableOn f s

Next tactic:
have hbelow : BddBelow (f '' s) := ⟨f a, fun x ⟨y, hy, hyx⟩ => hyx ▸ hmono ha.1 hy (ha.2 hy)⟩
theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b : X}
    (ha : IsLeast s a) (hb : IsGreatest s b) (hs : μ s ≠ ∞) (h's : MeasurableSet s) :
    IntegrableOn f s μ := by
  borelize E
  obtain rfl | _ := s.eq_empty_or_nonempty
  · exact integrableOn_empty
  have hbelow : BddBelow (f '' s) := ⟨f a, fun x ⟨y, hy, hyx⟩ => hyx ▸ hmono ha.1 hy (ha.2 hy)⟩
  

Proof state:
case inr
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹¹ : MeasurableSpace X
inst✝¹⁰ : TopologicalSpace X
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
hs : ↑↑μ s ≠ ⊤
h's : MeasurableSet s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
h✝ : Set.Nonempty s
hbelow : BddBelow (f '' s)
⊢ IntegrableOn f s

Next tactic:
have habove : BddAbove (f '' s) := ⟨f b, fun x ⟨y, hy, hyx⟩ => hyx ▸ hmono hy hb.1 (hb.2 hy)⟩
theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b : X}
    (ha : IsLeast s a) (hb : IsGreatest s b) (hs : μ s ≠ ∞) (h's : MeasurableSet s) :
    IntegrableOn f s μ := by
  borelize E
  obtain rfl | _ := s.eq_empty_or_nonempty
  · exact integrableOn_empty
  have hbelow : BddBelow (f '' s) := ⟨f a, fun x ⟨y, hy, hyx⟩ => hyx ▸ hmono ha.1 hy (ha.2 hy)⟩
  have habove : BddAbove (f '' s) := ⟨f b, fun x ⟨y, hy, hyx⟩ => hyx ▸ hmono hy hb.1 (hb.2 hy)⟩
  

Proof state:
case inr
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹¹ : MeasurableSpace X
inst✝¹⁰ : TopologicalSpace X
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
hs : ↑↑μ s ≠ ⊤
h's : MeasurableSet s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
h✝ : Set.Nonempty s
hbelow : BddBelow (f '' s)
habove : BddAbove (f '' s)
⊢ IntegrableOn f s

Next tactic:
have : IsBounded (f '' s) := Metric.isBounded_of_bddAbove_of_bddBelow habove hbelow
theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b : X}
    (ha : IsLeast s a) (hb : IsGreatest s b) (hs : μ s ≠ ∞) (h's : MeasurableSet s) :
    IntegrableOn f s μ := by
  borelize E
  obtain rfl | _ := s.eq_empty_or_nonempty
  · exact integrableOn_empty
  have hbelow : BddBelow (f '' s) := ⟨f a, fun x ⟨y, hy, hyx⟩ => hyx ▸ hmono ha.1 hy (ha.2 hy)⟩
  have habove : BddAbove (f '' s) := ⟨f b, fun x ⟨y, hy, hyx⟩ => hyx ▸ hmono hy hb.1 (hb.2 hy)⟩
  have : IsBounded (f '' s) := Metric.isBounded_of_bddAbove_of_bddBelow habove hbelow
  

Proof state:
case inr
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹¹ : MeasurableSpace X
inst✝¹⁰ : TopologicalSpace X
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
hs : ↑↑μ s ≠ ⊤
h's : MeasurableSet s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
h✝ : Set.Nonempty s
hbelow : BddBelow (f '' s)
habove : BddAbove (f '' s)
this : IsBounded (f '' s)
⊢ IntegrableOn f s

Next tactic:
rcases isBounded_iff_forall_norm_le.mp this with ⟨C, hC⟩
theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b : X}
    (ha : IsLeast s a) (hb : IsGreatest s b) (hs : μ s ≠ ∞) (h's : MeasurableSet s) :
    IntegrableOn f s μ := by
  borelize E
  obtain rfl | _ := s.eq_empty_or_nonempty
  · exact integrableOn_empty
  have hbelow : BddBelow (f '' s) := ⟨f a, fun x ⟨y, hy, hyx⟩ => hyx ▸ hmono ha.1 hy (ha.2 hy)⟩
  have habove : BddAbove (f '' s) := ⟨f b, fun x ⟨y, hy, hyx⟩ => hyx ▸ hmono hy hb.1 (hb.2 hy)⟩
  have : IsBounded (f '' s) := Metric.isBounded_of_bddAbove_of_bddBelow habove hbelow
  rcases isBounded_iff_forall_norm_le.mp this with ⟨C, hC⟩
  

Proof state:
case inr.intro
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹¹ : MeasurableSpace X
inst✝¹⁰ : TopologicalSpace X
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
hs : ↑↑μ s ≠ ⊤
h's : MeasurableSet s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
h✝ : Set.Nonempty s
hbelow : BddBelow (f '' s)
habove : BddAbove (f '' s)
this : IsBounded (f '' s)
C : ℝ
hC : ∀ x ∈ f '' s, ‖x‖ ≤ C
⊢ IntegrableOn f s

Next tactic:
have A : IntegrableOn (fun _ => C) s μ := by
    simp only [hs.lt_top, integrableOn_const, or_true_iff]
theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b : X}
    (ha : IsLeast s a) (hb : IsGreatest s b) (hs : μ s ≠ ∞) (h's : MeasurableSet s) :
    IntegrableOn f s μ := by
  borelize E
  obtain rfl | _ := s.eq_empty_or_nonempty
  · exact integrableOn_empty
  have hbelow : BddBelow (f '' s) := ⟨f a, fun x ⟨y, hy, hyx⟩ => hyx ▸ hmono ha.1 hy (ha.2 hy)⟩
  have habove : BddAbove (f '' s) := ⟨f b, fun x ⟨y, hy, hyx⟩ => hyx ▸ hmono hy hb.1 (hb.2 hy)⟩
  have : IsBounded (f '' s) := Metric.isBounded_of_bddAbove_of_bddBelow habove hbelow
  rcases isBounded_iff_forall_norm_le.mp this with ⟨C, hC⟩
  have A : IntegrableOn (fun _ => C) s μ := by
    

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹¹ : MeasurableSpace X
inst✝¹⁰ : TopologicalSpace X
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
hs : ↑↑μ s ≠ ⊤
h's : MeasurableSet s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
h✝ : Set.Nonempty s
hbelow : BddBelow (f '' s)
habove : BddAbove (f '' s)
this : IsBounded (f '' s)
C : ℝ
hC : ∀ x ∈ f '' s, ‖x‖ ≤ C
⊢ IntegrableOn (fun x => C) s

Next tactic:
simp only [hs.lt_top, integrableOn_const, or_true_iff]
theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b : X}
    (ha : IsLeast s a) (hb : IsGreatest s b) (hs : μ s ≠ ∞) (h's : MeasurableSet s) :
    IntegrableOn f s μ := by
  borelize E
  obtain rfl | _ := s.eq_empty_or_nonempty
  · exact integrableOn_empty
  have hbelow : BddBelow (f '' s) := ⟨f a, fun x ⟨y, hy, hyx⟩ => hyx ▸ hmono ha.1 hy (ha.2 hy)⟩
  have habove : BddAbove (f '' s) := ⟨f b, fun x ⟨y, hy, hyx⟩ => hyx ▸ hmono hy hb.1 (hb.2 hy)⟩
  have : IsBounded (f '' s) := Metric.isBounded_of_bddAbove_of_bddBelow habove hbelow
  rcases isBounded_iff_forall_norm_le.mp this with ⟨C, hC⟩
  have A : IntegrableOn (fun _ => C) s μ := by
    simp only [hs.lt_top, integrableOn_const, or_true_iff]
  

Proof state:
case inr.intro
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹¹ : MeasurableSpace X
inst✝¹⁰ : TopologicalSpace X
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
hs : ↑↑μ s ≠ ⊤
h's : MeasurableSet s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
h✝ : Set.Nonempty s
hbelow : BddBelow (f '' s)
habove : BddAbove (f '' s)
this : IsBounded (f '' s)
C : ℝ
hC : ∀ x ∈ f '' s, ‖x‖ ≤ C
A : IntegrableOn (fun x => C) s
⊢ IntegrableOn f s

Next tactic:
refine'
    Integrable.mono' A (aemeasurable_restrict_of_monotoneOn h's hmono).aestronglyMeasurable
      ((ae_restrict_iff' h's).mpr <| ae_of_all _ fun y hy => hC (f y) (mem_image_of_mem f hy))
theorem MonotoneOn.integrableOn_isCompact [IsFiniteMeasureOnCompacts μ] (hs : IsCompact s)
    (hmono : MonotoneOn f s) : IntegrableOn f s μ := by
  

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹² : MeasurableSpace X
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace Y
inst✝⁹ : TopologicalSpace Y
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
inst✝ : IsFiniteMeasureOnCompacts μ
hs : IsCompact s
hmono : MonotoneOn f s
⊢ IntegrableOn f s

Next tactic:
obtain rfl | h := s.eq_empty_or_nonempty
theorem MonotoneOn.integrableOn_isCompact [IsFiniteMeasureOnCompacts μ] (hs : IsCompact s)
    (hmono : MonotoneOn f s) : IntegrableOn f s μ := by
  obtain rfl | h := s.eq_empty_or_nonempty
  · 

Proof state:
case inl
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹² : MeasurableSpace X
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace Y
inst✝⁹ : TopologicalSpace Y
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
f g : X → E
μ : Measure X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
inst✝ : IsFiniteMeasureOnCompacts μ
hs : IsCompact ∅
hmono : MonotoneOn f ∅
⊢ IntegrableOn f ∅

Next tactic:
exact integrableOn_empty
theorem MonotoneOn.integrableOn_isCompact [IsFiniteMeasureOnCompacts μ] (hs : IsCompact s)
    (hmono : MonotoneOn f s) : IntegrableOn f s μ := by
  obtain rfl | h := s.eq_empty_or_nonempty
  · exact integrableOn_empty
  · 

Proof state:
case inr
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹² : MeasurableSpace X
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace Y
inst✝⁹ : TopologicalSpace Y
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
inst✝ : IsFiniteMeasureOnCompacts μ
hs : IsCompact s
hmono : MonotoneOn f s
h : Set.Nonempty s
⊢ IntegrableOn f s

Next tactic:
exact
      hmono.integrableOn_of_measure_ne_top (hs.isLeast_sInf h) (hs.isGreatest_sSup h)
        hs.measure_lt_top.ne hs.measurableSet
theorem Monotone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hmono : Monotone f) :
    LocallyIntegrable f μ := by
  

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹² : MeasurableSpace X
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace Y
inst✝⁹ : TopologicalSpace Y
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
inst✝ : IsLocallyFiniteMeasure μ
hmono : Monotone f
⊢ LocallyIntegrable f

Next tactic:
intro x
theorem Monotone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hmono : Monotone f) :
    LocallyIntegrable f μ := by
  intro x
  

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹² : MeasurableSpace X
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace Y
inst✝⁹ : TopologicalSpace Y
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
inst✝ : IsLocallyFiniteMeasure μ
hmono : Monotone f
x : X
⊢ IntegrableAtFilter f (𝓝 x)

Next tactic:
rcases μ.finiteAt_nhds x with ⟨U, hU, h'U⟩
theorem Monotone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hmono : Monotone f) :
    LocallyIntegrable f μ := by
  intro x
  rcases μ.finiteAt_nhds x with ⟨U, hU, h'U⟩
  

Proof state:
case intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹² : MeasurableSpace X
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace Y
inst✝⁹ : TopologicalSpace Y
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
inst✝ : IsLocallyFiniteMeasure μ
hmono : Monotone f
x : X
U : Set X
hU : U ∈ 𝓝 x
h'U : ↑↑μ U < ⊤
⊢ IntegrableAtFilter f (𝓝 x)

Next tactic:
obtain ⟨a, b, xab, hab, abU⟩ : ∃ a b : X, x ∈ Icc a b ∧ Icc a b ∈ 𝓝 x ∧ Icc a b ⊆ U
theorem Monotone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hmono : Monotone f) :
    LocallyIntegrable f μ := by
  intro x
  rcases μ.finiteAt_nhds x with ⟨U, hU, h'U⟩
  obtain ⟨a, b, xab, hab, abU⟩ : ∃ a b : X, x ∈ Icc a b ∧ Icc a b ∈ 𝓝 x ∧ Icc a b ⊆ U
  

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹² : MeasurableSpace X
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace Y
inst✝⁹ : TopologicalSpace Y
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
inst✝ : IsLocallyFiniteMeasure μ
hmono : Monotone f
x : X
U : Set X
hU : U ∈ 𝓝 x
h'U : ↑↑μ U < ⊤
⊢ ∃ a b, x ∈ Icc a b ∧ Icc a b ∈ 𝓝 x ∧ Icc a b ⊆ U
case intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹² : MeasurableSpace X
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace Y
inst✝⁹ : TopologicalSpace Y
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
inst✝ : IsLocallyFiniteMeasure μ
hmono : Monotone f
x : X
U : Set X
hU : U ∈ 𝓝 x
h'U : ↑↑μ U < ⊤
a b : X
xab : x ∈ Icc a b
hab : Icc a b ∈ 𝓝 x
abU : Icc a b ⊆ U
⊢ IntegrableAtFilter f (𝓝 x)

Next tactic:
exact exists_Icc_mem_subset_of_mem_nhds hU
theorem Monotone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hmono : Monotone f) :
    LocallyIntegrable f μ := by
  intro x
  rcases μ.finiteAt_nhds x with ⟨U, hU, h'U⟩
  obtain ⟨a, b, xab, hab, abU⟩ : ∃ a b : X, x ∈ Icc a b ∧ Icc a b ∈ 𝓝 x ∧ Icc a b ⊆ U
  exact exists_Icc_mem_subset_of_mem_nhds hU
  

Proof state:
case intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹² : MeasurableSpace X
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace Y
inst✝⁹ : TopologicalSpace Y
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
inst✝ : IsLocallyFiniteMeasure μ
hmono : Monotone f
x : X
U : Set X
hU : U ∈ 𝓝 x
h'U : ↑↑μ U < ⊤
a b : X
xab : x ∈ Icc a b
hab : Icc a b ∈ 𝓝 x
abU : Icc a b ⊆ U
⊢ IntegrableAtFilter f (𝓝 x)

Next tactic:
have ab : a ≤ b := xab.1.trans xab.2
theorem Monotone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hmono : Monotone f) :
    LocallyIntegrable f μ := by
  intro x
  rcases μ.finiteAt_nhds x with ⟨U, hU, h'U⟩
  obtain ⟨a, b, xab, hab, abU⟩ : ∃ a b : X, x ∈ Icc a b ∧ Icc a b ∈ 𝓝 x ∧ Icc a b ⊆ U
  exact exists_Icc_mem_subset_of_mem_nhds hU
  have ab : a ≤ b := xab.1.trans xab.2
  

Proof state:
case intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹² : MeasurableSpace X
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace Y
inst✝⁹ : TopologicalSpace Y
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
inst✝ : IsLocallyFiniteMeasure μ
hmono : Monotone f
x : X
U : Set X
hU : U ∈ 𝓝 x
h'U : ↑↑μ U < ⊤
a b : X
xab : x ∈ Icc a b
hab : Icc a b ∈ 𝓝 x
abU : Icc a b ⊆ U
ab : a ≤ b
⊢ IntegrableAtFilter f (𝓝 x)

Next tactic:
refine' ⟨Icc a b, hab, _⟩
theorem Monotone.locallyIntegrable [IsLocallyFiniteMeasure μ] (hmono : Monotone f) :
    LocallyIntegrable f μ := by
  intro x
  rcases μ.finiteAt_nhds x with ⟨U, hU, h'U⟩
  obtain ⟨a, b, xab, hab, abU⟩ : ∃ a b : X, x ∈ Icc a b ∧ Icc a b ∈ 𝓝 x ∧ Icc a b ⊆ U
  exact exists_Icc_mem_subset_of_mem_nhds hU
  have ab : a ≤ b := xab.1.trans xab.2
  refine' ⟨Icc a b, hab, _⟩
  

Proof state:
case intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹² : MeasurableSpace X
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace Y
inst✝⁹ : TopologicalSpace Y
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
f g : X → E
μ : Measure X
s : Set X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
inst✝ : IsLocallyFiniteMeasure μ
hmono : Monotone f
x : X
U : Set X
hU : U ∈ 𝓝 x
h'U : ↑↑μ U < ⊤
a b : X
xab : x ∈ Icc a b
hab : Icc a b ∈ 𝓝 x
abU : Icc a b ⊆ U
ab : a ≤ b
⊢ IntegrableOn f (Icc a b)

Next tactic:
exact
    (hmono.monotoneOn _).integrableOn_of_measure_ne_top (isLeast_Icc ab) (isGreatest_Icc ab)
      ((measure_mono abU).trans_lt h'U).ne measurableSet_Icc
theorem IntegrableOn.mul_continuousOn_of_subset (hg : IntegrableOn g A μ) (hg' : ContinuousOn g' K)
    (hA : MeasurableSet A) (hK : IsCompact K) (hAK : A ⊆ K) :
    IntegrableOn (fun x => g x * g' x) A μ := by
  

Proof state:
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f g✝ : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : IntegrableOn g A
hg' : ContinuousOn g' K
hA : MeasurableSet A
hK : IsCompact K
hAK : A ⊆ K
⊢ IntegrableOn (fun x => g x * g' x) A

Next tactic:
rcases IsCompact.exists_bound_of_continuousOn hK hg' with ⟨C, hC⟩
theorem IntegrableOn.mul_continuousOn_of_subset (hg : IntegrableOn g A μ) (hg' : ContinuousOn g' K)
    (hA : MeasurableSet A) (hK : IsCompact K) (hAK : A ⊆ K) :
    IntegrableOn (fun x => g x * g' x) A μ := by
  rcases IsCompact.exists_bound_of_continuousOn hK hg' with ⟨C, hC⟩
  

Proof state:
case intro
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
f g✝ : X → E
μ : Measure X
s : Set X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : IntegrableOn g A
hg' : ContinuousOn g' K
hA : MeasurableSet A
hK : IsCompact K
hAK : A ⊆ K
C : ℝ
hC : ∀ x ∈ K, ‖g' x‖ ≤ C
⊢ IntegrableOn (fun x => g x * g' x) A

Next tactic:
rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg ⊢