ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasurableSpace.countablePartitionSet_eq_iff
lemma countablePartitionSet_eq_iff {n : ℕ} (a : α) {s : Set α} (hs : s ∈ countablePartition α n) :
    countablePartitionSet n a = s ↔ a ∈ s 
Arguments: {α : Type u_1} [m : MeasurableSpace α] [h : MeasurableSpace.CountablyGenerated α] {n : ℕ} (a : α) {s : Set α} (hs : s ∈ MeasurableSpace.countablePartition α n)
Goal: MeasurableSpace.countablePartitionSet n a = s ↔ a ∈ s
-- MeasurableSpace.countablePartitionSet_of_mem
lemma countablePartitionSet_of_mem {n : ℕ} {a : α} {s : Set α} (hs : s ∈ countablePartition α n)
    (ha : a ∈ s) :
    countablePartitionSet n a = s 
Arguments: {α : Type u_1} [m : MeasurableSpace α] [h : MeasurableSpace.CountablyGenerated α] {n : ℕ} {a : α} {s : Set α} (hs : s ∈ MeasurableSpace.countablePartition α n) (ha : a ∈ s)
Goal: MeasurableSpace.countablePartitionSet n a = s
-- MeasurableSpace.measurableSet_countablePartitionSet
lemma measurableSet_countablePartitionSet (n : ℕ) (a : α) :
    MeasurableSet (countablePartitionSet n a) 
Arguments: {α : Type u_1} [m : MeasurableSpace α] [h : MeasurableSpace.CountablyGenerated α] (n : ℕ) (a : α)
Goal: MeasurableSet (MeasurableSpace.countablePartitionSet n a)
-- MeasurableSpace.CountableOrCountablyGenerated.countableOrCountablyGenerated
class CountableOrCountablyGenerated (α β : Type*) [MeasurableSpace α] [MeasurableSpace β] : Prop :=
  countableOrCountablyGenerated : Countable α ∨ MeasurableSpace.CountablyGenerated β
Arguments: {α : Type u_3} {β : Type u_4} [MeasurableSpace α] [MeasurableSpace β] [self : MeasurableSpace.CountableOrCountablyGenerated α β]
Goal: Countable α ∨ MeasurableSpace.CountablyGenerated β
-- MeasureTheory.rnDeriv_ae_eq_condexp
theorem rnDeriv_ae_eq_condexp {hm : m ≤ m0} [hμm : SigmaFinite (μ.trim hm)] {f : α → ℝ}
    (hf : Integrable f μ) :
    SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m] 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {hm : m ≤ m0} [hμm : MeasureTheory.SigmaFinite (μ.trim hm)] {f : α → ℝ} (hf : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) =ᵐ[μ] μ[f|m]
-- MeasureTheory.eLpNorm_one_condexp_le_eLpNorm
theorem eLpNorm_one_condexp_le_eLpNorm (f : α → ℝ) : eLpNorm (μ[f|m]) 1 μ ≤ eLpNorm f 1 μ 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ℝ)
Goal: MeasureTheory.eLpNorm (μ[f|m]) 1 μ ≤ MeasureTheory.eLpNorm f 1 μ
-- MeasureTheory.integral_abs_condexp_le
theorem integral_abs_condexp_le (f : α → ℝ) : ∫ x, |(μ[f|m]) x| ∂μ ≤ ∫ x, |f x| ∂μ 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ℝ)
Goal: ∫ (x : α), |(μ[f|m]) x| ∂μ ≤ ∫ (x : α), |f x| ∂μ
-- MeasureTheory.setIntegral_abs_condexp_le
theorem setIntegral_abs_condexp_le {s : Set α} (hs : MeasurableSet[m] s) (f : α → ℝ) :
    ∫ x in s, |(μ[f|m]) x| ∂μ ≤ ∫ x in s, |f x| ∂μ 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) (f : α → ℝ)
Goal: ∫ (x : α) in s, |(μ[f|m]) x| ∂μ ≤ ∫ (x : α) in s, |f x| ∂μ
-- MeasureTheory.ae_bdd_condexp_of_ae_bdd
theorem ae_bdd_condexp_of_ae_bdd {R : ℝ≥0} {f : α → ℝ} (hbdd : ∀ᵐ x ∂μ, |f x| ≤ R) :
    ∀ᵐ x ∂μ, |(μ[f|m]) x| ≤ R 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {R : ℝ≥0} {f : α → ℝ} (hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R)
Goal: ∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑R
-- MeasureTheory.Integrable.uniformIntegrable_condexp
theorem Integrable.uniformIntegrable_condexp {ι : Type*} [IsFiniteMeasure μ] {g : α → ℝ}
    (hint : Integrable g μ) {ℱ : ι → MeasurableSpace α} (hℱ : ∀ i, ℱ i ≤ m0) :
    UniformIntegrable (fun i => μ[g|ℱ i]) 1 μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_2} [MeasureTheory.IsFiniteMeasure μ] {g : α → ℝ} (hint : MeasureTheory.Integrable g μ) {ℱ : ι → MeasurableSpace α} (hℱ : ∀ (i : ι), ℱ i ≤ m0)
Goal: MeasureTheory.UniformIntegrable (fun i => μ[g|ℱ i]) 1 μ
-- MeasureTheory.condexp_stronglyMeasurable_simpleFunc_mul
theorem condexp_stronglyMeasurable_simpleFunc_mul (hm : m ≤ m0) (f : @SimpleFunc α m ℝ) {g : α → ℝ}
    (hg : Integrable g μ) : μ[(f * g : α → ℝ)|m] =ᵐ[μ] f * μ[g|m] 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) (f : MeasureTheory.SimpleFunc α ℝ) {g : α → ℝ} (hg : MeasureTheory.Integrable g μ)
Goal: μ[↑f * g|m] =ᵐ[μ] ↑f * μ[g|m]
-- MeasureTheory.condexp_stronglyMeasurable_mul_of_bound
theorem condexp_stronglyMeasurable_mul_of_bound (hm : m ≤ m0) [IsFiniteMeasure μ] {f g : α → ℝ}
    (hf : StronglyMeasurable[m] f) (hg : Integrable g μ) (c : ℝ) (hf_bound : ∀ᵐ x ∂μ, ‖f x‖ ≤ c) :
    μ[f * g|m] =ᵐ[μ] f * μ[g|m] 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) [MeasureTheory.IsFiniteMeasure μ] {f g : α → ℝ} (hf : MeasureTheory.StronglyMeasurable f) (hg : MeasureTheory.Integrable g μ) (c : ℝ) (hf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c)
Goal: μ[f * g|m] =ᵐ[μ] f * μ[g|m]
-- MeasureTheory.condexp_stronglyMeasurable_mul_of_bound₀
theorem condexp_stronglyMeasurable_mul_of_bound₀ (hm : m ≤ m0) [IsFiniteMeasure μ] {f g : α → ℝ}
    (hf : AEStronglyMeasurable' m f μ) (hg : Integrable g μ) (c : ℝ)
    (hf_bound : ∀ᵐ x ∂μ, ‖f x‖ ≤ c) : μ[f * g|m] =ᵐ[μ] f * μ[g|m] 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) [MeasureTheory.IsFiniteMeasure μ] {f g : α → ℝ} (hf : MeasureTheory.AEStronglyMeasurable' m f μ) (hg : MeasureTheory.Integrable g μ) (c : ℝ) (hf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c)
Goal: μ[f * g|m] =ᵐ[μ] f * μ[g|m]
-- MeasureTheory.condexp_stronglyMeasurable_mul
theorem condexp_stronglyMeasurable_mul {f g : α → ℝ} (hf : StronglyMeasurable[m] f)
    (hfg : Integrable (f * g) μ) (hg : Integrable g μ) : μ[f * g|m] =ᵐ[μ] f * μ[g|m] 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ} (hf : MeasureTheory.StronglyMeasurable f) (hfg : MeasureTheory.Integrable (f * g) μ) (hg : MeasureTheory.Integrable g μ)
Goal: μ[f * g|m] =ᵐ[μ] f * μ[g|m]
-- MeasureTheory.condexp_stronglyMeasurable_mul₀
theorem condexp_stronglyMeasurable_mul₀ {f g : α → ℝ} (hf : AEStronglyMeasurable' m f μ)
    (hfg : Integrable (f * g) μ) (hg : Integrable g μ) : μ[f * g|m] =ᵐ[μ] f * μ[g|m] 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ} (hf : MeasureTheory.AEStronglyMeasurable' m f μ) (hfg : MeasureTheory.Integrable (f * g) μ) (hg : MeasureTheory.Integrable g μ)
Goal: μ[f * g|m] =ᵐ[μ] f * μ[g|m]
-- MeasureTheory.OuterMeasure.ofFunction_apply
theorem ofFunction_apply (s : Set α) :
    OuterMeasure.ofFunction m m_empty s = ⨅ (t : ℕ → Set α) (_ : s ⊆ iUnion t), ∑' n, m (t n) 
Arguments: {α : Type u_1} (m : Set α → ℝ≥0∞) (m_empty : m ∅ = 0) (s : Set α)
Goal: (MeasureTheory.OuterMeasure.ofFunction m m_empty) s = ⨅ t, ⨅ (_ : s ⊆ Set.iUnion t), ∑' (n : ℕ), m (t n)
-- MeasureTheory.OuterMeasure.ofFunction_le
theorem ofFunction_le (s : Set α) : OuterMeasure.ofFunction m m_empty s ≤ m s 
Arguments: {α : Type u_1} {m : Set α → ℝ≥0∞} {m_empty : m ∅ = 0} (s : Set α)
Goal: (MeasureTheory.OuterMeasure.ofFunction m m_empty) s ≤ m s
-- MeasureTheory.OuterMeasure.ofFunction_eq
theorem ofFunction_eq (s : Set α) (m_mono : ∀ ⦃t : Set α⦄, s ⊆ t → m s ≤ m t)
    (m_subadd : ∀ s : ℕ → Set α, m (⋃ i, s i) ≤ ∑' i, m (s i)) :
    OuterMeasure.ofFunction m m_empty s = m s 
Arguments: {α : Type u_1} {m : Set α → ℝ≥0∞} {m_empty : m ∅ = 0} (s : Set α) (m_mono : ∀ ⦃t : Set α⦄, s ⊆ t → m s ≤ m t) (m_subadd : ∀ (s : ℕ → Set α), m (⋃ i, s i) ≤ ∑' (i : ℕ), m (s i))
Goal: (MeasureTheory.OuterMeasure.ofFunction m m_empty) s = m s
-- MeasureTheory.OuterMeasure.le_ofFunction
theorem le_ofFunction {μ : OuterMeasure α} :
    μ ≤ OuterMeasure.ofFunction m m_empty ↔ ∀ s, μ s ≤ m s 
Arguments: {α : Type u_1} {m : Set α → ℝ≥0∞} {m_empty : m ∅ = 0} {μ : MeasureTheory.OuterMeasure α}
Goal: μ ≤ MeasureTheory.OuterMeasure.ofFunction m m_empty ↔ ∀ (s : Set α), μ s ≤ m s
-- MeasureTheory.OuterMeasure.isGreatest_ofFunction
theorem isGreatest_ofFunction :
    IsGreatest { μ : OuterMeasure α | ∀ s, μ s ≤ m s } (OuterMeasure.ofFunction m m_empty) 
Arguments: {α : Type u_1} {m : Set α → ℝ≥0∞} {m_empty : m ∅ = 0}
Goal: IsGreatest {μ | ∀ (s : Set α), μ s ≤ m s} (MeasureTheory.OuterMeasure.ofFunction m m_empty)
-- MeasureTheory.OuterMeasure.ofFunction_eq_sSup
theorem ofFunction_eq_sSup : OuterMeasure.ofFunction m m_empty = sSup { μ | ∀ s, μ s ≤ m s } 
Arguments: {α : Type u_1} {m : Set α → ℝ≥0∞} {m_empty : m ∅ = 0}
Goal: MeasureTheory.OuterMeasure.ofFunction m m_empty = sSup {μ | ∀ (s : Set α), μ s ≤ m s}
-- MeasureTheory.OuterMeasure.ofFunction_union_of_top_of_nonempty_inter
theorem ofFunction_union_of_top_of_nonempty_inter {s t : Set α}
    (h : ∀ u, (s ∩ u).Nonempty → (t ∩ u).Nonempty → m u = ∞) :
    OuterMeasure.ofFunction m m_empty (s ∪ t) =
      OuterMeasure.ofFunction m m_empty s + OuterMeasure.ofFunction m m_empty t 
Arguments: {α : Type u_1} {m : Set α → ℝ≥0∞} {m_empty : m ∅ = 0} {s t : Set α} (h : ∀ (u : Set α), (s ∩ u).Nonempty → (t ∩ u).Nonempty → m u = ⊤)
Goal: (MeasureTheory.OuterMeasure.ofFunction m m_empty) (s ∪ t) =
  (MeasureTheory.OuterMeasure.ofFunction m m_empty) s + (MeasureTheory.OuterMeasure.ofFunction m m_empty) t
-- MeasureTheory.OuterMeasure.comap_ofFunction
theorem comap_ofFunction {β} (f : β → α) (h : Monotone m ∨ Surjective f) :
    comap f (OuterMeasure.ofFunction m m_empty) =
      OuterMeasure.ofFunction (fun s => m (f '' s)) (by simp; simp [m_empty]) 
Arguments: {α : Type u_1} {m : Set α → ℝ≥0∞} {m_empty : m ∅ = 0} {β : Type u_2} (f : β → α) (h : Monotone m ∨ Function.Surjective f)
Goal: (MeasureTheory.OuterMeasure.comap f) (MeasureTheory.OuterMeasure.ofFunction m m_empty) =
  MeasureTheory.OuterMeasure.ofFunction (fun s => m (f '' s)) ⋯
-- MeasureTheory.OuterMeasure.map_ofFunction_le
theorem map_ofFunction_le {β} (f : α → β) :
    map f (OuterMeasure.ofFunction m m_empty) ≤
      OuterMeasure.ofFunction (fun s => m (f ⁻¹' s)) m_empty 
Arguments: {α : Type u_1} {m : Set α → ℝ≥0∞} {m_empty : m ∅ = 0} {β : Type u_2} (f : α → β)
Goal: (MeasureTheory.OuterMeasure.map f) (MeasureTheory.OuterMeasure.ofFunction m m_empty) ≤
  MeasureTheory.OuterMeasure.ofFunction (fun s => m (f ⁻¹' s)) m_empty
-- MeasureTheory.OuterMeasure.map_ofFunction
theorem map_ofFunction {β} {f : α → β} (hf : Injective f) :
    map f (OuterMeasure.ofFunction m m_empty) =
      OuterMeasure.ofFunction (fun s => m (f ⁻¹' s)) m_empty 
Arguments: {α : Type u_1} {m : Set α → ℝ≥0∞} {m_empty : m ∅ = 0} {β : Type u_2} {f : α → β} (hf : Function.Injective f)
Goal: (MeasureTheory.OuterMeasure.map f) (MeasureTheory.OuterMeasure.ofFunction m m_empty) =
  MeasureTheory.OuterMeasure.ofFunction (fun s => m (f ⁻¹' s)) m_empty
-- MeasureTheory.OuterMeasure.restrict_ofFunction
theorem restrict_ofFunction (s : Set α) (hm : Monotone m) :
    restrict s (OuterMeasure.ofFunction m m_empty) =
      OuterMeasure.ofFunction (fun t => m (t ∩ s)) (by simp; simp [m_empty]) 
Arguments: {α : Type u_1} {m : Set α → ℝ≥0∞} {m_empty : m ∅ = 0} (s : Set α) (hm : Monotone m)
Goal: (MeasureTheory.OuterMeasure.restrict s) (MeasureTheory.OuterMeasure.ofFunction m m_empty) =
  MeasureTheory.OuterMeasure.ofFunction (fun t => m (t ∩ s)) ⋯
-- MeasureTheory.OuterMeasure.smul_ofFunction
theorem smul_ofFunction {c : ℝ≥0∞} (hc : c ≠ ∞) : c • OuterMeasure.ofFunction m m_empty =
    OuterMeasure.ofFunction (c • m) (by simp [m_empty]) 
Arguments: {α : Type u_1} {m : Set α → ℝ≥0∞} {m_empty : m ∅ = 0} {c : ℝ≥0∞} (hc : c ≠ ⊤)
Goal: c • MeasureTheory.OuterMeasure.ofFunction m m_empty = MeasureTheory.OuterMeasure.ofFunction (c • m) ⋯
-- MeasureTheory.OuterMeasure.boundedBy_le
theorem boundedBy_le (s : Set α) : boundedBy m s ≤ m s 
Arguments: {α : Type u_1} {m : Set α → ℝ≥0∞} (s : Set α)
Goal: (MeasureTheory.OuterMeasure.boundedBy m) s ≤ m s
-- MeasureTheory.OuterMeasure.boundedBy_eq_ofFunction
theorem boundedBy_eq_ofFunction (m_empty : m ∅ = 0) (s : Set α) :
    boundedBy m s = OuterMeasure.ofFunction m m_empty s 
Arguments: {α : Type u_1} {m : Set α → ℝ≥0∞} (m_empty : m ∅ = 0) (s : Set α)
Goal: (MeasureTheory.OuterMeasure.boundedBy m) s = (MeasureTheory.OuterMeasure.ofFunction m m_empty) s
-- MeasureTheory.OuterMeasure.boundedBy_apply
theorem boundedBy_apply (s : Set α) :
    boundedBy m s = ⨅ (t : ℕ → Set α) (_ : s ⊆ iUnion t),
                      ∑' n, ⨆ _ : (t n).Nonempty, m (t n) 
Arguments: {α : Type u_1} {m : Set α → ℝ≥0∞} (s : Set α)
Goal: (MeasureTheory.OuterMeasure.boundedBy m) s = ⨅ t, ⨅ (_ : s ⊆ Set.iUnion t), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), m (t n)
-- MeasureTheory.OuterMeasure.boundedBy_eq
theorem boundedBy_eq (s : Set α) (m_empty : m ∅ = 0) (m_mono : ∀ ⦃t : Set α⦄, s ⊆ t → m s ≤ m t)
    (m_subadd : ∀ s : ℕ → Set α, m (⋃ i, s i) ≤ ∑' i, m (s i)) : boundedBy m s = m s 
Arguments: {α : Type u_1} {m : Set α → ℝ≥0∞} (s : Set α) (m_empty : m ∅ = 0) (m_mono : ∀ ⦃t : Set α⦄, s ⊆ t → m s ≤ m t) (m_subadd : ∀ (s : ℕ → Set α), m (⋃ i, s i) ≤ ∑' (i : ℕ), m (s i))
Goal: (MeasureTheory.OuterMeasure.boundedBy m) s = m s
-- MeasureTheory.OuterMeasure.boundedBy_eq_self
@[simp]
theorem boundedBy_eq_self (m : OuterMeasure α) : boundedBy m = m 
Arguments: {α : Type u_1} (m : MeasureTheory.OuterMeasure α)
Goal: MeasureTheory.OuterMeasure.boundedBy ⇑m = m
-- MeasureTheory.OuterMeasure.le_boundedBy
theorem le_boundedBy {μ : OuterMeasure α} : μ ≤ boundedBy m ↔ ∀ s, μ s ≤ m s 
Arguments: {α : Type u_1} {m : Set α → ℝ≥0∞} {μ : MeasureTheory.OuterMeasure α}
Goal: μ ≤ MeasureTheory.OuterMeasure.boundedBy m ↔ ∀ (s : Set α), μ s ≤ m s
-- MeasureTheory.OuterMeasure.le_boundedBy'
theorem le_boundedBy' {μ : OuterMeasure α} :
    μ ≤ boundedBy m ↔ ∀ s : Set α, s.Nonempty → μ s ≤ m s 
Arguments: {α : Type u_1} {m : Set α → ℝ≥0∞} {μ : MeasureTheory.OuterMeasure α}
Goal: μ ≤ MeasureTheory.OuterMeasure.boundedBy m ↔ ∀ (s : Set α), s.Nonempty → μ s ≤ m s
-- MeasureTheory.OuterMeasure.boundedBy_top
@[simp]
theorem boundedBy_top : boundedBy (⊤ : Set α → ℝ≥0∞) = ⊤ 
Arguments: {α : Type u_1}
Goal: MeasureTheory.OuterMeasure.boundedBy ⊤ = ⊤
-- MeasureTheory.OuterMeasure.boundedBy_zero
@[simp]
theorem boundedBy_zero : boundedBy (0 : Set α → ℝ≥0∞) = 0 
Arguments: {α : Type u_1}
Goal: MeasureTheory.OuterMeasure.boundedBy 0 = 0
-- MeasureTheory.OuterMeasure.smul_boundedBy
theorem smul_boundedBy {c : ℝ≥0∞} (hc : c ≠ ∞) : c • boundedBy m = boundedBy (c • m) 
Arguments: {α : Type u_1} {m : Set α → ℝ≥0∞} {c : ℝ≥0∞} (hc : c ≠ ⊤)
Goal: c • MeasureTheory.OuterMeasure.boundedBy m = MeasureTheory.OuterMeasure.boundedBy (c • m)
-- MeasureTheory.OuterMeasure.comap_boundedBy
theorem comap_boundedBy {β} (f : β → α)
    (h : (Monotone fun s : { s : Set α // s.Nonempty } => m s) ∨ Surjective f) :
    comap f (boundedBy m) = boundedBy fun s => m (f '' s) 
Arguments: {α : Type u_1} {m : Set α → ℝ≥0∞} {β : Type u_2} (f : β → α) (h : (Monotone fun s => m ↑s) ∨ Function.Surjective f)
Goal: (MeasureTheory.OuterMeasure.comap f) (MeasureTheory.OuterMeasure.boundedBy m) =
  MeasureTheory.OuterMeasure.boundedBy fun s => m (f '' s)
-- MeasureTheory.OuterMeasure.boundedBy_union_of_top_of_nonempty_inter
theorem boundedBy_union_of_top_of_nonempty_inter {s t : Set α}
    (h : ∀ u, (s ∩ u).Nonempty → (t ∩ u).Nonempty → m u = ∞) :
    boundedBy m (s ∪ t) = boundedBy m s + boundedBy m t 
Arguments: {α : Type u_1} {m : Set α → ℝ≥0∞} {s t : Set α} (h : ∀ (u : Set α), (s ∩ u).Nonempty → (t ∩ u).Nonempty → m u = ⊤)
Goal: (MeasureTheory.OuterMeasure.boundedBy m) (s ∪ t) =
  (MeasureTheory.OuterMeasure.boundedBy m) s + (MeasureTheory.OuterMeasure.boundedBy m) t
-- MeasureTheory.OuterMeasure.sInfGen_def
theorem sInfGen_def (m : Set (OuterMeasure α)) (t : Set α) :
    sInfGen m t = ⨅ (μ : OuterMeasure α) (_ : μ ∈ m), μ t 
Arguments: {α : Type u_1} (m : Set (MeasureTheory.OuterMeasure α)) (t : Set α)
Goal: MeasureTheory.OuterMeasure.sInfGen m t = ⨅ μ ∈ m, μ t
-- MeasureTheory.OuterMeasure.sInf_eq_boundedBy_sInfGen
theorem sInf_eq_boundedBy_sInfGen (m : Set (OuterMeasure α)) :
    sInf m = OuterMeasure.boundedBy (sInfGen m) 
Arguments: {α : Type u_1} (m : Set (MeasureTheory.OuterMeasure α))
Goal: sInf m = MeasureTheory.OuterMeasure.boundedBy (MeasureTheory.OuterMeasure.sInfGen m)
-- MeasureTheory.OuterMeasure.iSup_sInfGen_nonempty
theorem iSup_sInfGen_nonempty {m : Set (OuterMeasure α)} (h : m.Nonempty) (t : Set α) :
    ⨆ _ : t.Nonempty, sInfGen m t = ⨅ (μ : OuterMeasure α) (_ : μ ∈ m), μ t 
Arguments: {α : Type u_1} {m : Set (MeasureTheory.OuterMeasure α)} (h : m.Nonempty) (t : Set α)
Goal: ⨆ (_ : t.Nonempty), MeasureTheory.OuterMeasure.sInfGen m t = ⨅ μ ∈ m, μ t
-- MeasureTheory.OuterMeasure.sInf_apply
theorem sInf_apply {m : Set (OuterMeasure α)} {s : Set α} (h : m.Nonempty) :
    sInf m s =
      ⨅ (t : ℕ → Set α) (_ : s ⊆ iUnion t), ∑' n, ⨅ (μ : OuterMeasure α) (_ : μ ∈ m), μ (t n) 
Arguments: {α : Type u_1} {m : Set (MeasureTheory.OuterMeasure α)} {s : Set α} (h : m.Nonempty)
Goal: (sInf m) s = ⨅ t, ⨅ (_ : s ⊆ Set.iUnion t), ∑' (n : ℕ), ⨅ μ ∈ m, μ (t n)
-- MeasureTheory.OuterMeasure.sInf_apply'
theorem sInf_apply' {m : Set (OuterMeasure α)} {s : Set α} (h : s.Nonempty) :
    sInf m s =
      ⨅ (t : ℕ → Set α) (_ : s ⊆ iUnion t), ∑' n, ⨅ (μ : OuterMeasure α) (_ : μ ∈ m), μ (t n) 
Arguments: {α : Type u_1} {m : Set (MeasureTheory.OuterMeasure α)} {s : Set α} (h : s.Nonempty)
Goal: (sInf m) s = ⨅ t, ⨅ (_ : s ⊆ Set.iUnion t), ∑' (n : ℕ), ⨅ μ ∈ m, μ (t n)
-- MeasureTheory.OuterMeasure.iInf_apply
theorem iInf_apply {ι} [Nonempty ι] (m : ι → OuterMeasure α) (s : Set α) :
    (⨅ i, m i) s = ⨅ (t : ℕ → Set α) (_ : s ⊆ iUnion t), ∑' n, ⨅ i, m i (t n) 
Arguments: {α : Type u_1} {ι : Sort u_2} [Nonempty ι] (m : ι → MeasureTheory.OuterMeasure α) (s : Set α)
Goal: (⨅ i, m i) s = ⨅ t, ⨅ (_ : s ⊆ Set.iUnion t), ∑' (n : ℕ), ⨅ i, (m i) (t n)
-- MeasureTheory.OuterMeasure.iInf_apply'
theorem iInf_apply' {ι} (m : ι → OuterMeasure α) {s : Set α} (hs : s.Nonempty) :
    (⨅ i, m i) s = ⨅ (t : ℕ → Set α) (_ : s ⊆ iUnion t), ∑' n, ⨅ i, m i (t n) 
Arguments: {α : Type u_1} {ι : Sort u_2} (m : ι → MeasureTheory.OuterMeasure α) {s : Set α} (hs : s.Nonempty)
Goal: (⨅ i, m i) s = ⨅ t, ⨅ (_ : s ⊆ Set.iUnion t), ∑' (n : ℕ), ⨅ i, (m i) (t n)
-- MeasureTheory.OuterMeasure.biInf_apply
theorem biInf_apply {ι} {I : Set ι} (hI : I.Nonempty) (m : ι → OuterMeasure α) (s : Set α) :
    (⨅ i ∈ I, m i) s = ⨅ (t : ℕ → Set α) (_ : s ⊆ iUnion t), ∑' n, ⨅ i ∈ I, m i (t n) 
Arguments: {α : Type u_1} {ι : Type u_2} {I : Set ι} (hI : I.Nonempty) (m : ι → MeasureTheory.OuterMeasure α) (s : Set α)
Goal: (⨅ i ∈ I, m i) s = ⨅ t, ⨅ (_ : s ⊆ Set.iUnion t), ∑' (n : ℕ), ⨅ i ∈ I, (m i) (t n)
-- MeasureTheory.OuterMeasure.biInf_apply'
theorem biInf_apply' {ι} (I : Set ι) (m : ι → OuterMeasure α) {s : Set α} (hs : s.Nonempty) :
    (⨅ i ∈ I, m i) s = ⨅ (t : ℕ → Set α) (_ : s ⊆ iUnion t), ∑' n, ⨅ i ∈ I, m i (t n) 
Arguments: {α : Type u_1} {ι : Type u_2} (I : Set ι) (m : ι → MeasureTheory.OuterMeasure α) {s : Set α} (hs : s.Nonempty)
Goal: (⨅ i ∈ I, m i) s = ⨅ t, ⨅ (_ : s ⊆ Set.iUnion t), ∑' (n : ℕ), ⨅ i ∈ I, (m i) (t n)
-- MeasureTheory.OuterMeasure.map_iInf_le
theorem map_iInf_le {ι β} (f : α → β) (m : ι → OuterMeasure α) :
    map f (⨅ i, m i) ≤ ⨅ i, map f (m i) 
Arguments: {α : Type u_1} {ι : Sort u_2} {β : Type u_3} (f : α → β) (m : ι → MeasureTheory.OuterMeasure α)
Goal: (MeasureTheory.OuterMeasure.map f) (⨅ i, m i) ≤ ⨅ i, (MeasureTheory.OuterMeasure.map f) (m i)
-- MeasureTheory.OuterMeasure.comap_iInf
theorem comap_iInf {ι β} (f : α → β) (m : ι → OuterMeasure β) :
    comap f (⨅ i, m i) = ⨅ i, comap f (m i) 
Arguments: {α : Type u_1} {ι : Sort u_2} {β : Type u_3} (f : α → β) (m : ι → MeasureTheory.OuterMeasure β)
Goal: (MeasureTheory.OuterMeasure.comap f) (⨅ i, m i) = ⨅ i, (MeasureTheory.OuterMeasure.comap f) (m i)
-- MeasureTheory.OuterMeasure.map_iInf
theorem map_iInf {ι β} {f : α → β} (hf : Injective f) (m : ι → OuterMeasure α) :
    map f (⨅ i, m i) = restrict (range f) (⨅ i, map f (m i)) 
Arguments: {α : Type u_1} {ι : Sort u_2} {β : Type u_3} {f : α → β} (hf : Function.Injective f) (m : ι → MeasureTheory.OuterMeasure α)
Goal: (MeasureTheory.OuterMeasure.map f) (⨅ i, m i) =
  (MeasureTheory.OuterMeasure.restrict (Set.range f)) (⨅ i, (MeasureTheory.OuterMeasure.map f) (m i))
-- MeasureTheory.OuterMeasure.map_iInf_comap
theorem map_iInf_comap {ι β} [Nonempty ι] {f : α → β} (m : ι → OuterMeasure β) :
    map f (⨅ i, comap f (m i)) = ⨅ i, map f (comap f (m i)) 
Arguments: {α : Type u_1} {ι : Sort u_2} {β : Type u_3} [Nonempty ι] {f : α → β} (m : ι → MeasureTheory.OuterMeasure β)
Goal: (MeasureTheory.OuterMeasure.map f) (⨅ i, (MeasureTheory.OuterMeasure.comap f) (m i)) =
  ⨅ i, (MeasureTheory.OuterMeasure.map f) ((MeasureTheory.OuterMeasure.comap f) (m i))
-- MeasureTheory.OuterMeasure.map_biInf_comap
theorem map_biInf_comap {ι β} {I : Set ι} (hI : I.Nonempty) {f : α → β} (m : ι → OuterMeasure β) :
    map f (⨅ i ∈ I, comap f (m i)) = ⨅ i ∈ I, map f (comap f (m i)) 
Arguments: {α : Type u_1} {ι : Type u_2} {β : Type u_3} {I : Set ι} (hI : I.Nonempty) {f : α → β} (m : ι → MeasureTheory.OuterMeasure β)
Goal: (MeasureTheory.OuterMeasure.map f) (⨅ i ∈ I, (MeasureTheory.OuterMeasure.comap f) (m i)) =
  ⨅ i ∈ I, (MeasureTheory.OuterMeasure.map f) ((MeasureTheory.OuterMeasure.comap f) (m i))
-- MeasureTheory.OuterMeasure.restrict_iInf_restrict
theorem restrict_iInf_restrict {ι} (s : Set α) (m : ι → OuterMeasure α) :
    restrict s (⨅ i, restrict s (m i)) = restrict s (⨅ i, m i) 
Arguments: {α : Type u_1} {ι : Sort u_2} (s : Set α) (m : ι → MeasureTheory.OuterMeasure α)
Goal: (MeasureTheory.OuterMeasure.restrict s) (⨅ i, (MeasureTheory.OuterMeasure.restrict s) (m i)) =
  (MeasureTheory.OuterMeasure.restrict s) (⨅ i, m i)
-- MeasureTheory.OuterMeasure.restrict_iInf
theorem restrict_iInf {ι} [Nonempty ι] (s : Set α) (m : ι → OuterMeasure α) :
    restrict s (⨅ i, m i) = ⨅ i, restrict s (m i) 
Arguments: {α : Type u_1} {ι : Sort u_2} [Nonempty ι] (s : Set α) (m : ι → MeasureTheory.OuterMeasure α)
Goal: (MeasureTheory.OuterMeasure.restrict s) (⨅ i, m i) = ⨅ i, (MeasureTheory.OuterMeasure.restrict s) (m i)
-- MeasureTheory.OuterMeasure.restrict_biInf
theorem restrict_biInf {ι} {I : Set ι} (hI : I.Nonempty) (s : Set α) (m : ι → OuterMeasure α) :
    restrict s (⨅ i ∈ I, m i) = ⨅ i ∈ I, restrict s (m i) 
Arguments: {α : Type u_1} {ι : Type u_2} {I : Set ι} (hI : I.Nonempty) (s : Set α) (m : ι → MeasureTheory.OuterMeasure α)
Goal: (MeasureTheory.OuterMeasure.restrict s) (⨅ i ∈ I, m i) = ⨅ i ∈ I, (MeasureTheory.OuterMeasure.restrict s) (m i)
-- MeasureTheory.OuterMeasure.restrict_sInf_eq_sInf_restrict
theorem restrict_sInf_eq_sInf_restrict (m : Set (OuterMeasure α)) {s : Set α} (hm : m.Nonempty) :
    restrict s (sInf m) = sInf (restrict s '' m) 
Arguments: {α : Type u_1} (m : Set (MeasureTheory.OuterMeasure α)) {s : Set α} (hm : m.Nonempty)
Goal: (MeasureTheory.OuterMeasure.restrict s) (sInf m) = sInf (⇑(MeasureTheory.OuterMeasure.restrict s) '' m)
-- MeasureTheory.extend_eq
theorem extend_eq {s : α} (h : P s) : extend m s = m s h 
Arguments: {α : Type u_1} {P : α → Prop} (m : (s : α) → P s → ℝ≥0∞) {s : α} (h : P s)
Goal: MeasureTheory.extend m s = m s h
-- MeasureTheory.extend_eq_top
theorem extend_eq_top {s : α} (h : ¬P s) : extend m s = ∞ 
Arguments: {α : Type u_1} {P : α → Prop} (m : (s : α) → P s → ℝ≥0∞) {s : α} (h : ¬P s)
Goal: MeasureTheory.extend m s = ⊤
-- MeasureTheory.smul_extend
theorem smul_extend {R} [Zero R] [SMulWithZero R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞]
    [NoZeroSMulDivisors R ℝ≥0∞] {c : R} (hc : c ≠ 0) :
    c • extend m = extend fun s h => c • m s h 
Arguments: {α : Type u_1} {P : α → Prop} (m : (s : α) → P s → ℝ≥0∞) {R : Type u_2} [Zero R] [SMulWithZero R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] [NoZeroSMulDivisors R ℝ≥0∞] {c : R} (hc : c ≠ 0)
Goal: c • MeasureTheory.extend m = MeasureTheory.extend fun s h => c • m s h
-- MeasureTheory.le_extend
theorem le_extend {s : α} (h : P s) : m s h ≤ extend m s 
Arguments: {α : Type u_1} {P : α → Prop} (m : (s : α) → P s → ℝ≥0∞) {s : α} (h : P s)
Goal: m s h ≤ MeasureTheory.extend m s