l3lab/ntp-mathlib huggingface dataset:
/-- A function which is continuous on a set `s` is almost everywhere strongly measurable with
respect to `μ.restrict s` when either the source space or the target space is second-countable. -/
theorem ContinuousOn.aestronglyMeasurable [TopologicalSpace α] [TopologicalSpace β]
    [h : SecondCountableTopologyEither α β] [OpensMeasurableSpace α] [PseudoMetrizableSpace β]
    {f : α → β} {s : Set α} {μ : Measure α} (hf : ContinuousOn f s) (hs : MeasurableSet s) :
    AEStronglyMeasurable f (μ.restrict s) := by
  borelize β
  refine'
    aestronglyMeasurable_iff_aemeasurable_separable.2
      ⟨hf.aemeasurable hs, f '' s, _,
        mem_of_superset (self_mem_ae_restrict hs) (subset_preimage_image _ _)⟩
  cases h.out
  · let f' : s → β := s.restrict f
    have A : Continuous f' := continuousOn_iff_continuous_restrict.1 hf
    have B : IsSeparable (univ : Set s) := isSeparable_of_separableSpace _
    convert IsSeparable.image B A using 1
    ext x
    simp
  · 

Proof state:
case inr
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
h : SecondCountableTopologyEither α β
inst✝¹ : OpensMeasurableSpace α
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : MeasurableSet s
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
h✝ : SecondCountableTopology β
⊢ IsSeparable (f '' s)

Next tactic:
exact isSeparable_of_separableSpace _
/-- A function which is continuous on a compact set `s` is almost everywhere strongly measurable
with respect to `μ.restrict s`. -/
theorem ContinuousOn.aestronglyMeasurable_of_isCompact [TopologicalSpace α] [OpensMeasurableSpace α]
    [TopologicalSpace β] [PseudoMetrizableSpace β] {f : α → β} {s : Set α} {μ : Measure α}
    (hf : ContinuousOn f s) (hs : IsCompact s) (h's : MeasurableSet s) :
    AEStronglyMeasurable f (μ.restrict s) := by
  

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : IsCompact s
h's : MeasurableSet s
⊢ AEStronglyMeasurable f (Measure.restrict μ s)

Next tactic:
letI := pseudoMetrizableSpacePseudoMetric β
/-- A function which is continuous on a compact set `s` is almost everywhere strongly measurable
with respect to `μ.restrict s`. -/
theorem ContinuousOn.aestronglyMeasurable_of_isCompact [TopologicalSpace α] [OpensMeasurableSpace α]
    [TopologicalSpace β] [PseudoMetrizableSpace β] {f : α → β} {s : Set α} {μ : Measure α}
    (hf : ContinuousOn f s) (hs : IsCompact s) (h's : MeasurableSet s) :
    AEStronglyMeasurable f (μ.restrict s) := by
  letI := pseudoMetrizableSpacePseudoMetric β
  

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : IsCompact s
h's : MeasurableSet s
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
⊢ AEStronglyMeasurable f (Measure.restrict μ s)

Next tactic:
borelize β
/-- A function which is continuous on a compact set `s` is almost everywhere strongly measurable
with respect to `μ.restrict s`. -/
theorem ContinuousOn.aestronglyMeasurable_of_isCompact [TopologicalSpace α] [OpensMeasurableSpace α]
    [TopologicalSpace β] [PseudoMetrizableSpace β] {f : α → β} {s : Set α} {μ : Measure α}
    (hf : ContinuousOn f s) (hs : IsCompact s) (h's : MeasurableSet s) :
    AEStronglyMeasurable f (μ.restrict s) := by
  letI := pseudoMetrizableSpacePseudoMetric β
  borelize β
  

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : IsCompact s
h's : MeasurableSet s
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ AEStronglyMeasurable f (Measure.restrict μ s)

Next tactic:
rw [aestronglyMeasurable_iff_aemeasurable_separable]
/-- A function which is continuous on a compact set `s` is almost everywhere strongly measurable
with respect to `μ.restrict s`. -/
theorem ContinuousOn.aestronglyMeasurable_of_isCompact [TopologicalSpace α] [OpensMeasurableSpace α]
    [TopologicalSpace β] [PseudoMetrizableSpace β] {f : α → β} {s : Set α} {μ : Measure α}
    (hf : ContinuousOn f s) (hs : IsCompact s) (h's : MeasurableSet s) :
    AEStronglyMeasurable f (μ.restrict s) := by
  letI := pseudoMetrizableSpacePseudoMetric β
  borelize β
  rw [aestronglyMeasurable_iff_aemeasurable_separable]
  

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : IsCompact s
h's : MeasurableSet s
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ AEMeasurable f ∧ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂Measure.restrict μ s, f x ∈ t

Next tactic:
refine' ⟨hf.aemeasurable h's, f '' s, _, _⟩
/-- A function which is continuous on a compact set `s` is almost everywhere strongly measurable
with respect to `μ.restrict s`. -/
theorem ContinuousOn.aestronglyMeasurable_of_isCompact [TopologicalSpace α] [OpensMeasurableSpace α]
    [TopologicalSpace β] [PseudoMetrizableSpace β] {f : α → β} {s : Set α} {μ : Measure α}
    (hf : ContinuousOn f s) (hs : IsCompact s) (h's : MeasurableSet s) :
    AEStronglyMeasurable f (μ.restrict s) := by
  letI := pseudoMetrizableSpacePseudoMetric β
  borelize β
  rw [aestronglyMeasurable_iff_aemeasurable_separable]
  refine' ⟨hf.aemeasurable h's, f '' s, _, _⟩
  · 

Proof state:
case refine'_1
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : IsCompact s
h's : MeasurableSet s
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ IsSeparable (f '' s)

Next tactic:
exact (hs.image_of_continuousOn hf).isSeparable
/-- A function which is continuous on a compact set `s` is almost everywhere strongly measurable
with respect to `μ.restrict s`. -/
theorem ContinuousOn.aestronglyMeasurable_of_isCompact [TopologicalSpace α] [OpensMeasurableSpace α]
    [TopologicalSpace β] [PseudoMetrizableSpace β] {f : α → β} {s : Set α} {μ : Measure α}
    (hf : ContinuousOn f s) (hs : IsCompact s) (h's : MeasurableSet s) :
    AEStronglyMeasurable f (μ.restrict s) := by
  letI := pseudoMetrizableSpacePseudoMetric β
  borelize β
  rw [aestronglyMeasurable_iff_aemeasurable_separable]
  refine' ⟨hf.aemeasurable h's, f '' s, _, _⟩
  · exact (hs.image_of_continuousOn hf).isSeparable
  · 

Proof state:
case refine'_2
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : IsCompact s
h's : MeasurableSet s
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ ∀ᵐ (x : α) ∂Measure.restrict μ s, f x ∈ f '' s

Next tactic:
exact mem_of_superset (self_mem_ae_restrict h's) (subset_preimage_image _ _)
theorem Continuous.integrableAt_nhds [TopologicalSpace α] [SecondCountableTopologyEither α E]
    [OpensMeasurableSpace α] {μ : Measure α} [IsLocallyFiniteMeasure μ] {f : α → E}
    (hf : Continuous f) (a : α) : IntegrableAtFilter f (𝓝 a) μ := by
  

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopologyEither α E
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : Continuous f
a : α
⊢ IntegrableAtFilter f (𝓝 a)

Next tactic:
rw [← nhdsWithin_univ]
theorem Continuous.integrableAt_nhds [TopologicalSpace α] [SecondCountableTopologyEither α E]
    [OpensMeasurableSpace α] {μ : Measure α} [IsLocallyFiniteMeasure μ] {f : α → E}
    (hf : Continuous f) (a : α) : IntegrableAtFilter f (𝓝 a) μ := by
  rw [← nhdsWithin_univ]
  

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopologyEither α E
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : Continuous f
a : α
⊢ IntegrableAtFilter f (𝓝[univ] a)

Next tactic:
exact hf.continuousOn.integrableAt_nhdsWithin MeasurableSet.univ (mem_univ a)
theorem integrableOn_Icc_iff_integrableOn_Ioc' (ha : μ {a} ≠ ∞) :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioc a b) μ := by
  

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
ha : ↑↑μ {a} ≠ ⊤
⊢ IntegrableOn f (Icc a b) ↔ IntegrableOn f (Ioc a b)

Next tactic:
by_cases hab : a ≤ b
theorem integrableOn_Icc_iff_integrableOn_Ioc' (ha : μ {a} ≠ ∞) :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioc a b) μ := by
  by_cases hab : a ≤ b
  · 

Proof state:
case pos
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
ha : ↑↑μ {a} ≠ ⊤
hab : a ≤ b
⊢ IntegrableOn f (Icc a b) ↔ IntegrableOn f (Ioc a b)

Next tactic:
rw [← Ioc_union_left hab, integrableOn_union,
      eq_true (integrableOn_singleton_iff.mpr <| Or.inr ha.lt_top), and_true_iff]
theorem integrableOn_Icc_iff_integrableOn_Ioc' (ha : μ {a} ≠ ∞) :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioc a b) μ := by
  by_cases hab : a ≤ b
  · rw [← Ioc_union_left hab, integrableOn_union,
      eq_true (integrableOn_singleton_iff.mpr <| Or.inr ha.lt_top), and_true_iff]
  · 

Proof state:
case neg
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
ha : ↑↑μ {a} ≠ ⊤
hab : ¬a ≤ b
⊢ IntegrableOn f (Icc a b) ↔ IntegrableOn f (Ioc a b)

Next tactic:
rw [Icc_eq_empty hab, Ioc_eq_empty]
theorem integrableOn_Icc_iff_integrableOn_Ioc' (ha : μ {a} ≠ ∞) :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioc a b) μ := by
  by_cases hab : a ≤ b
  · rw [← Ioc_union_left hab, integrableOn_union,
      eq_true (integrableOn_singleton_iff.mpr <| Or.inr ha.lt_top), and_true_iff]
  · rw [Icc_eq_empty hab, Ioc_eq_empty]
    

Proof state:
case neg
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
ha : ↑↑μ {a} ≠ ⊤
hab : ¬a ≤ b
⊢ ¬a < b

Next tactic:
contrapose! hab
theorem integrableOn_Icc_iff_integrableOn_Ioc' (ha : μ {a} ≠ ∞) :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioc a b) μ := by
  by_cases hab : a ≤ b
  · rw [← Ioc_union_left hab, integrableOn_union,
      eq_true (integrableOn_singleton_iff.mpr <| Or.inr ha.lt_top), and_true_iff]
  · rw [Icc_eq_empty hab, Ioc_eq_empty]
    contrapose! hab
    

Proof state:
case neg
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
ha : ↑↑μ {a} ≠ ⊤
hab : a < b
⊢ a ≤ b

Next tactic:
exact hab.le
theorem integrableOn_Icc_iff_integrableOn_Ico' (hb : μ {b} ≠ ∞) :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ico a b) μ := by
  

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
hb : ↑↑μ {b} ≠ ⊤
⊢ IntegrableOn f (Icc a b) ↔ IntegrableOn f (Ico a b)

Next tactic:
by_cases hab : a ≤ b
theorem integrableOn_Icc_iff_integrableOn_Ico' (hb : μ {b} ≠ ∞) :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ico a b) μ := by
  by_cases hab : a ≤ b
  · 

Proof state:
case pos
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
hb : ↑↑μ {b} ≠ ⊤
hab : a ≤ b
⊢ IntegrableOn f (Icc a b) ↔ IntegrableOn f (Ico a b)

Next tactic:
rw [← Ico_union_right hab, integrableOn_union,
      eq_true (integrableOn_singleton_iff.mpr <| Or.inr hb.lt_top), and_true_iff]
theorem integrableOn_Icc_iff_integrableOn_Ico' (hb : μ {b} ≠ ∞) :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ico a b) μ := by
  by_cases hab : a ≤ b
  · rw [← Ico_union_right hab, integrableOn_union,
      eq_true (integrableOn_singleton_iff.mpr <| Or.inr hb.lt_top), and_true_iff]
  · 

Proof state:
case neg
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
hb : ↑↑μ {b} ≠ ⊤
hab : ¬a ≤ b
⊢ IntegrableOn f (Icc a b) ↔ IntegrableOn f (Ico a b)

Next tactic:
rw [Icc_eq_empty hab, Ico_eq_empty]
theorem integrableOn_Icc_iff_integrableOn_Ico' (hb : μ {b} ≠ ∞) :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ico a b) μ := by
  by_cases hab : a ≤ b
  · rw [← Ico_union_right hab, integrableOn_union,
      eq_true (integrableOn_singleton_iff.mpr <| Or.inr hb.lt_top), and_true_iff]
  · rw [Icc_eq_empty hab, Ico_eq_empty]
    

Proof state:
case neg
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
hb : ↑↑μ {b} ≠ ⊤
hab : ¬a ≤ b
⊢ ¬a < b

Next tactic:
contrapose! hab
theorem integrableOn_Icc_iff_integrableOn_Ico' (hb : μ {b} ≠ ∞) :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ico a b) μ := by
  by_cases hab : a ≤ b
  · rw [← Ico_union_right hab, integrableOn_union,
      eq_true (integrableOn_singleton_iff.mpr <| Or.inr hb.lt_top), and_true_iff]
  · rw [Icc_eq_empty hab, Ico_eq_empty]
    contrapose! hab
    

Proof state:
case neg
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
hb : ↑↑μ {b} ≠ ⊤
hab : a < b
⊢ a ≤ b

Next tactic:
exact hab.le
theorem integrableOn_Ico_iff_integrableOn_Ioo' (ha : μ {a} ≠ ∞) :
    IntegrableOn f (Ico a b) μ ↔ IntegrableOn f (Ioo a b) μ := by
  

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
ha : ↑↑μ {a} ≠ ⊤
⊢ IntegrableOn f (Ico a b) ↔ IntegrableOn f (Ioo a b)

Next tactic:
by_cases hab : a < b
theorem integrableOn_Ico_iff_integrableOn_Ioo' (ha : μ {a} ≠ ∞) :
    IntegrableOn f (Ico a b) μ ↔ IntegrableOn f (Ioo a b) μ := by
  by_cases hab : a < b
  · 

Proof state:
case pos
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
ha : ↑↑μ {a} ≠ ⊤
hab : a < b
⊢ IntegrableOn f (Ico a b) ↔ IntegrableOn f (Ioo a b)

Next tactic:
rw [← Ioo_union_left hab, integrableOn_union,
      eq_true (integrableOn_singleton_iff.mpr <| Or.inr ha.lt_top), and_true_iff]
theorem integrableOn_Ico_iff_integrableOn_Ioo' (ha : μ {a} ≠ ∞) :
    IntegrableOn f (Ico a b) μ ↔ IntegrableOn f (Ioo a b) μ := by
  by_cases hab : a < b
  · rw [← Ioo_union_left hab, integrableOn_union,
      eq_true (integrableOn_singleton_iff.mpr <| Or.inr ha.lt_top), and_true_iff]
  · 

Proof state:
case neg
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
ha : ↑↑μ {a} ≠ ⊤
hab : ¬a < b
⊢ IntegrableOn f (Ico a b) ↔ IntegrableOn f (Ioo a b)

Next tactic:
rw [Ioo_eq_empty hab, Ico_eq_empty hab]
theorem integrableOn_Ioc_iff_integrableOn_Ioo' (hb : μ {b} ≠ ∞) :
    IntegrableOn f (Ioc a b) μ ↔ IntegrableOn f (Ioo a b) μ := by
  

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
hb : ↑↑μ {b} ≠ ⊤
⊢ IntegrableOn f (Ioc a b) ↔ IntegrableOn f (Ioo a b)

Next tactic:
by_cases hab : a < b
theorem integrableOn_Ioc_iff_integrableOn_Ioo' (hb : μ {b} ≠ ∞) :
    IntegrableOn f (Ioc a b) μ ↔ IntegrableOn f (Ioo a b) μ := by
  by_cases hab : a < b
  · 

Proof state:
case pos
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
hb : ↑↑μ {b} ≠ ⊤
hab : a < b
⊢ IntegrableOn f (Ioc a b) ↔ IntegrableOn f (Ioo a b)

Next tactic:
rw [← Ioo_union_right hab, integrableOn_union,
      eq_true (integrableOn_singleton_iff.mpr <| Or.inr hb.lt_top), and_true_iff]
theorem integrableOn_Ioc_iff_integrableOn_Ioo' (hb : μ {b} ≠ ∞) :
    IntegrableOn f (Ioc a b) μ ↔ IntegrableOn f (Ioo a b) μ := by
  by_cases hab : a < b
  · rw [← Ioo_union_right hab, integrableOn_union,
      eq_true (integrableOn_singleton_iff.mpr <| Or.inr hb.lt_top), and_true_iff]
  · 

Proof state:
case neg
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
hb : ↑↑μ {b} ≠ ⊤
hab : ¬a < b
⊢ IntegrableOn f (Ioc a b) ↔ IntegrableOn f (Ioo a b)

Next tactic:
rw [Ioo_eq_empty hab, Ioc_eq_empty hab]
theorem integrableOn_Icc_iff_integrableOn_Ioo' (ha : μ {a} ≠ ∞) (hb : μ {b} ≠ ∞) :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioo a b) μ := by
  

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
ha : ↑↑μ {a} ≠ ⊤
hb : ↑↑μ {b} ≠ ⊤
⊢ IntegrableOn f (Icc a b) ↔ IntegrableOn f (Ioo a b)

Next tactic:
rw [integrableOn_Icc_iff_integrableOn_Ioc' ha, integrableOn_Ioc_iff_integrableOn_Ioo' hb]
theorem integrableOn_Ici_iff_integrableOn_Ioi' (hb : μ {b} ≠ ∞) :
    IntegrableOn f (Ici b) μ ↔ IntegrableOn f (Ioi b) μ := by
  

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
hb : ↑↑μ {b} ≠ ⊤
⊢ IntegrableOn f (Ici b) ↔ IntegrableOn f (Ioi b)

Next tactic:
rw [← Ioi_union_left, integrableOn_union,
    eq_true (integrableOn_singleton_iff.mpr <| Or.inr hb.lt_top), and_true_iff]
theorem integrableOn_Iic_iff_integrableOn_Iio' (hb : μ {b} ≠ ∞) :
    IntegrableOn f (Iic b) μ ↔ IntegrableOn f (Iio b) μ := by
  

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder α
inst✝ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
hb : ↑↑μ {b} ≠ ⊤
⊢ IntegrableOn f (Iic b) ↔ IntegrableOn f (Iio b)

Next tactic:
rw [← Iio_union_right, integrableOn_union,
    eq_true (integrableOn_singleton_iff.mpr <| Or.inr hb.lt_top), and_true_iff]
theorem integrableOn_Icc_iff_integrableOn_Ioc :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioc a b) μ :=
  integrableOn_Icc_iff_integrableOn_Ioc' (by 

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : PartialOrder α
inst✝¹ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
inst✝ : NoAtoms μ
⊢ ↑↑μ {a} ≠ ⊤

Next tactic:
rw [measure_singleton]
theorem integrableOn_Icc_iff_integrableOn_Ioc :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioc a b) μ :=
  integrableOn_Icc_iff_integrableOn_Ioc' (by rw [measure_singleton]; 

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : PartialOrder α
inst✝¹ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
inst✝ : NoAtoms μ
⊢ 0 ≠ ⊤

Next tactic:
exact ENNReal.zero_ne_top
theorem integrableOn_Icc_iff_integrableOn_Ico :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ico a b) μ :=
  integrableOn_Icc_iff_integrableOn_Ico' (by 

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : PartialOrder α
inst✝¹ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
inst✝ : NoAtoms μ
⊢ ↑↑μ {b} ≠ ⊤

Next tactic:
rw [measure_singleton]
theorem integrableOn_Icc_iff_integrableOn_Ico :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ico a b) μ :=
  integrableOn_Icc_iff_integrableOn_Ico' (by rw [measure_singleton]; 

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : PartialOrder α
inst✝¹ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
inst✝ : NoAtoms μ
⊢ 0 ≠ ⊤

Next tactic:
exact ENNReal.zero_ne_top
theorem integrableOn_Ico_iff_integrableOn_Ioo :
    IntegrableOn f (Ico a b) μ ↔ IntegrableOn f (Ioo a b) μ :=
  integrableOn_Ico_iff_integrableOn_Ioo' (by 

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : PartialOrder α
inst✝¹ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
inst✝ : NoAtoms μ
⊢ ↑↑μ {a} ≠ ⊤

Next tactic:
rw [measure_singleton]
theorem integrableOn_Ico_iff_integrableOn_Ioo :
    IntegrableOn f (Ico a b) μ ↔ IntegrableOn f (Ioo a b) μ :=
  integrableOn_Ico_iff_integrableOn_Ioo' (by rw [measure_singleton]; 

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : PartialOrder α
inst✝¹ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
inst✝ : NoAtoms μ
⊢ 0 ≠ ⊤

Next tactic:
exact ENNReal.zero_ne_top
theorem integrableOn_Ioc_iff_integrableOn_Ioo :
    IntegrableOn f (Ioc a b) μ ↔ IntegrableOn f (Ioo a b) μ :=
  integrableOn_Ioc_iff_integrableOn_Ioo' (by 

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : PartialOrder α
inst✝¹ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
inst✝ : NoAtoms μ
⊢ ↑↑μ {b} ≠ ⊤

Next tactic:
rw [measure_singleton]
theorem integrableOn_Ioc_iff_integrableOn_Ioo :
    IntegrableOn f (Ioc a b) μ ↔ IntegrableOn f (Ioo a b) μ :=
  integrableOn_Ioc_iff_integrableOn_Ioo' (by rw [measure_singleton]; 

Proof state:
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : PartialOrder α
inst✝¹ : MeasurableSingletonClass α
f : α → E
μ : Measure α
a b : α
inst✝ : NoAtoms μ
⊢ 0 ≠ ⊤

Next tactic:
exact ENNReal.zero_ne_top