ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.StronglyMeasurable.norm
@[measurability]
protected theorem norm {_ : MeasurableSpace α} {β : Type*} [SeminormedAddCommGroup β] {f : α → β}
    (hf : StronglyMeasurable f) : StronglyMeasurable fun x => ‖f x‖ 
Arguments: {α : Type u_1}
Goal: ∀ {x : MeasurableSpace α} {β : Type u_5} [inst : SeminormedAddCommGroup β] {f : α → β},
  MeasureTheory.StronglyMeasurable f → MeasureTheory.StronglyMeasurable fun x => ‖f x‖
-- MeasureTheory.StronglyMeasurable.nnnorm
@[measurability]
protected theorem nnnorm {_ : MeasurableSpace α} {β : Type*} [SeminormedAddCommGroup β] {f : α → β}
    (hf : StronglyMeasurable f) : StronglyMeasurable fun x => ‖f x‖₊ 
Arguments: {α : Type u_1}
Goal: ∀ {x : MeasurableSpace α} {β : Type u_5} [inst : SeminormedAddCommGroup β] {f : α → β},
  MeasureTheory.StronglyMeasurable f → MeasureTheory.StronglyMeasurable fun x => ‖f x‖₊
-- MeasureTheory.StronglyMeasurable.ennnorm
@[measurability]
protected theorem ennnorm {_ : MeasurableSpace α} {β : Type*} [SeminormedAddCommGroup β]
    {f : α → β} (hf : StronglyMeasurable f) : Measurable fun a => (‖f a‖₊ : ℝ≥0∞) 
Arguments: {α : Type u_1}
Goal: ∀ {x : MeasurableSpace α} {β : Type u_5} [inst : SeminormedAddCommGroup β] {f : α → β},
  MeasureTheory.StronglyMeasurable f → Measurable fun a => ↑‖f a‖₊
-- MeasureTheory.StronglyMeasurable.real_toNNReal
@[measurability]
protected theorem real_toNNReal {_ : MeasurableSpace α} {f : α → ℝ} (hf : StronglyMeasurable f) :
    StronglyMeasurable fun x => (f x).toNNReal 
Arguments: {α : Type u_1}
Goal: ∀ {x : MeasurableSpace α} {f : α → ℝ},
  MeasureTheory.StronglyMeasurable f → MeasureTheory.StronglyMeasurable fun x => (f x).toNNReal
-- MeasureTheory.StronglyMeasurable.measurableSet_eq_fun
theorem measurableSet_eq_fun {m : MeasurableSpace α} {E} [TopologicalSpace E] [MetrizableSpace E]
    {f g : α → E} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
    MeasurableSet { x | f x = g x } 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {E : Type u_5} [TopologicalSpace E] [TopologicalSpace.MetrizableSpace E] {f g : α → E} (hf : MeasureTheory.StronglyMeasurable f) (hg : MeasureTheory.StronglyMeasurable g)
Goal: MeasurableSet {x | f x = g x}
-- MeasureTheory.StronglyMeasurable.measurableSet_lt
theorem measurableSet_lt {m : MeasurableSpace α} [TopologicalSpace β] [LinearOrder β]
    [OrderClosedTopology β] [PseudoMetrizableSpace β] {f g : α → β} (hf : StronglyMeasurable f)
    (hg : StronglyMeasurable g) : MeasurableSet { a | f a < g a } 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [TopologicalSpace β] [LinearOrder β] [OrderClosedTopology β] [TopologicalSpace.PseudoMetrizableSpace β] {f g : α → β} (hf : MeasureTheory.StronglyMeasurable f) (hg : MeasureTheory.StronglyMeasurable g)
Goal: MeasurableSet {a | f a < g a}
-- MeasureTheory.StronglyMeasurable.measurableSet_le
theorem measurableSet_le {m : MeasurableSpace α} [TopologicalSpace β] [Preorder β]
    [OrderClosedTopology β] [PseudoMetrizableSpace β] {f g : α → β} (hf : StronglyMeasurable f)
    (hg : StronglyMeasurable g) : MeasurableSet { a | f a ≤ g a } 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [TopologicalSpace β] [Preorder β] [OrderClosedTopology β] [TopologicalSpace.PseudoMetrizableSpace β] {f g : α → β} (hf : MeasureTheory.StronglyMeasurable f) (hg : MeasureTheory.StronglyMeasurable g)
Goal: MeasurableSet {a | f a ≤ g a}
-- MeasureTheory.StronglyMeasurable.stronglyMeasurable_in_set
theorem stronglyMeasurable_in_set {m : MeasurableSpace α} [TopologicalSpace β] [Zero β] {s : Set α}
    {f : α → β} (hs : MeasurableSet s) (hf : StronglyMeasurable f)
    (hf_zero : ∀ x, x ∉ s → f x = 0) :
    ∃ fs : ℕ → α →ₛ β,
      (∀ x, Tendsto (fun n => fs n x) atTop (𝓝 (f x))) ∧ ∀ x ∉ s, ∀ n, fs n x = 0 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [TopologicalSpace β] [Zero β] {s : Set α} {f : α → β} (hs : MeasurableSet s) (hf : MeasureTheory.StronglyMeasurable f) (hf_zero : ∀ x ∉ s, f x = 0)
Goal: ∃ fs, (∀ (x : α), Filter.Tendsto (fun n => ↑(fs n) x) Filter.atTop (𝓝 (f x))) ∧ ∀ x ∉ s, ∀ (n : ℕ), ↑(fs n) x = 0
-- MeasureTheory.StronglyMeasurable.stronglyMeasurable_of_measurableSpace_le_on
theorem stronglyMeasurable_of_measurableSpace_le_on {α E} {m m₂ : MeasurableSpace α}
    [TopologicalSpace E] [Zero E] {s : Set α} {f : α → E} (hs_m : MeasurableSet[m] s)
    (hs : ∀ t, MeasurableSet[m] (s ∩ t) → MeasurableSet[m₂] (s ∩ t))
    (hf : StronglyMeasurable[m] f) (hf_zero : ∀ x ∉ s, f x = 0) :
    StronglyMeasurable[m₂] f 
Arguments: {α : Type u_5} {E : Type u_6} {m m₂ : MeasurableSpace α} [TopologicalSpace E] [Zero E] {s : Set α} {f : α → E} (hs_m : MeasurableSet s) (hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)) (hf : MeasureTheory.StronglyMeasurable f) (hf_zero : ∀ x ∉ s, f x = 0)
Goal: MeasureTheory.StronglyMeasurable f
-- MeasureTheory.StronglyMeasurable.exists_spanning_measurableSet_norm_le
theorem exists_spanning_measurableSet_norm_le [SeminormedAddCommGroup β] {m m0 : MeasurableSpace α}
    (hm : m ≤ m0) (hf : StronglyMeasurable[m] f) (μ : Measure α) [SigmaFinite (μ.trim hm)] :
    ∃ s : ℕ → Set α,
      (∀ n, MeasurableSet[m] (s n) ∧ μ (s n) < ∞ ∧ ∀ x ∈ s n, ‖f x‖ ≤ n) ∧
      ⋃ i, s i = Set.univ 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [SeminormedAddCommGroup β] {m m0 : MeasurableSpace α} (hm : m ≤ m0) (hf : MeasureTheory.StronglyMeasurable f) (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite (μ.trim hm)]
Goal: ∃ s, (∀ (n : ℕ), MeasurableSet (s n) ∧ μ (s n) < ⊤ ∧ ∀ x ∈ s n, ‖f x‖ ≤ ↑n) ∧ ⋃ i, s i = Set.univ
-- MeasureTheory.finStronglyMeasurable_zero
theorem finStronglyMeasurable_zero {α β} {m : MeasurableSpace α} {μ : Measure α} [Zero β]
    [TopologicalSpace β] : FinStronglyMeasurable (0 : α → β) μ 
Arguments: {α : Type u_5} {β : Type u_6} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Zero β] [TopologicalSpace β]
Goal: MeasureTheory.FinStronglyMeasurable 0 μ
-- MeasureTheory.FinStronglyMeasurable.aefinStronglyMeasurable
theorem aefinStronglyMeasurable [Zero β] [TopologicalSpace β] (hf : FinStronglyMeasurable f μ) :
    AEFinStronglyMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → β} [Zero β] [TopologicalSpace β] (hf : MeasureTheory.FinStronglyMeasurable f μ)
Goal: MeasureTheory.AEFinStronglyMeasurable f μ
-- MeasureTheory.FinStronglyMeasurable.fin_support_approx
protected theorem fin_support_approx : ∀ n, μ (support (hf.approx n)) < ∞ 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → β} [Zero β] [TopologicalSpace β] (hf : MeasureTheory.FinStronglyMeasurable f μ) (n : ℕ)
Goal: μ (Function.support ↑(hf.approx n)) < ⊤
-- MeasureTheory.FinStronglyMeasurable.tendsto_approx
protected theorem tendsto_approx : ∀ x, Tendsto (fun n => hf.approx n x) atTop (𝓝 (f x)) 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → β} [Zero β] [TopologicalSpace β] (hf : MeasureTheory.FinStronglyMeasurable f μ) (x : α)
Goal: Filter.Tendsto (fun n => ↑(hf.approx n) x) Filter.atTop (𝓝 (f x))
-- MeasureTheory.FinStronglyMeasurable.stronglyMeasurable
@[aesop 5% apply (rule_sets := [Measurable])]
protected theorem stronglyMeasurable [Zero β] [TopologicalSpace β]
    (hf : FinStronglyMeasurable f μ) : StronglyMeasurable f 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → β} [Zero β] [TopologicalSpace β] (hf : MeasureTheory.FinStronglyMeasurable f μ)
Goal: MeasureTheory.StronglyMeasurable f
-- MeasureTheory.FinStronglyMeasurable.exists_set_sigmaFinite
theorem exists_set_sigmaFinite [Zero β] [TopologicalSpace β] [T2Space β]
    (hf : FinStronglyMeasurable f μ) :
    ∃ t, MeasurableSet t ∧ (∀ x ∈ tᶜ, f x = 0) ∧ SigmaFinite (μ.restrict t) 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → β} [Zero β] [TopologicalSpace β] [T2Space β] (hf : MeasureTheory.FinStronglyMeasurable f μ)
Goal: ∃ t, MeasurableSet t ∧ (∀ x ∈ tᶜ, f x = 0) ∧ MeasureTheory.SigmaFinite (μ.restrict t)
-- MeasureTheory.FinStronglyMeasurable.measurable
protected theorem measurable [Zero β] [TopologicalSpace β] [PseudoMetrizableSpace β]
    [MeasurableSpace β] [BorelSpace β] (hf : FinStronglyMeasurable f μ) : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → β} [Zero β] [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β] (hf : MeasureTheory.FinStronglyMeasurable f μ)
Goal: Measurable f
-- MeasureTheory.FinStronglyMeasurable.mul
@[aesop safe 20 (rule_sets := [Measurable])]
protected theorem mul [MonoidWithZero β] [ContinuousMul β] (hf : FinStronglyMeasurable f μ)
    (hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f * g) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → β} [TopologicalSpace β] [MonoidWithZero β] [ContinuousMul β] (hf : MeasureTheory.FinStronglyMeasurable f μ) (hg : MeasureTheory.FinStronglyMeasurable g μ)
Goal: MeasureTheory.FinStronglyMeasurable (f * g) μ
-- MeasureTheory.FinStronglyMeasurable.add
@[aesop safe 20 (rule_sets := [Measurable])]
protected theorem add [AddMonoid β] [ContinuousAdd β] (hf : FinStronglyMeasurable f μ)
    (hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f + g) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → β} [TopologicalSpace β] [AddMonoid β] [ContinuousAdd β] (hf : MeasureTheory.FinStronglyMeasurable f μ) (hg : MeasureTheory.FinStronglyMeasurable g μ)
Goal: MeasureTheory.FinStronglyMeasurable (f + g) μ
-- MeasureTheory.FinStronglyMeasurable.neg
@[measurability]
protected theorem neg [AddGroup β] [TopologicalAddGroup β] (hf : FinStronglyMeasurable f μ) :
    FinStronglyMeasurable (-f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → β} [TopologicalSpace β] [AddGroup β] [TopologicalAddGroup β] (hf : MeasureTheory.FinStronglyMeasurable f μ)
Goal: MeasureTheory.FinStronglyMeasurable (-f) μ
-- MeasureTheory.FinStronglyMeasurable.sub
@[measurability]
protected theorem sub [AddGroup β] [ContinuousSub β] (hf : FinStronglyMeasurable f μ)
    (hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f - g) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → β} [TopologicalSpace β] [AddGroup β] [ContinuousSub β] (hf : MeasureTheory.FinStronglyMeasurable f μ) (hg : MeasureTheory.FinStronglyMeasurable g μ)
Goal: MeasureTheory.FinStronglyMeasurable (f - g) μ
-- MeasureTheory.FinStronglyMeasurable.const_smul
@[measurability]
protected theorem const_smul {𝕜} [TopologicalSpace 𝕜] [AddMonoid β] [Monoid 𝕜]
    [DistribMulAction 𝕜 β] [ContinuousSMul 𝕜 β] (hf : FinStronglyMeasurable f μ) (c : 𝕜) :
    FinStronglyMeasurable (c • f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → β} [TopologicalSpace β] {𝕜 : Type u_5} [TopologicalSpace 𝕜] [AddMonoid β] [Monoid 𝕜] [DistribMulAction 𝕜 β] [ContinuousSMul 𝕜 β] (hf : MeasureTheory.FinStronglyMeasurable f μ) (c : 𝕜)
Goal: MeasureTheory.FinStronglyMeasurable (c • f) μ
-- MeasureTheory.FinStronglyMeasurable.sup
@[aesop safe 20 (rule_sets := [Measurable])]
protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : FinStronglyMeasurable f μ)
    (hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f ⊔ g) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → β} [TopologicalSpace β] [Zero β] [SemilatticeSup β] [ContinuousSup β] (hf : MeasureTheory.FinStronglyMeasurable f μ) (hg : MeasureTheory.FinStronglyMeasurable g μ)
Goal: MeasureTheory.FinStronglyMeasurable (f ⊔ g) μ
-- MeasureTheory.FinStronglyMeasurable.inf
@[aesop safe 20 (rule_sets := [Measurable])]
protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : FinStronglyMeasurable f μ)
    (hg : FinStronglyMeasurable g μ) : FinStronglyMeasurable (f ⊓ g) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → β} [TopologicalSpace β] [Zero β] [SemilatticeInf β] [ContinuousInf β] (hf : MeasureTheory.FinStronglyMeasurable f μ) (hg : MeasureTheory.FinStronglyMeasurable g μ)
Goal: MeasureTheory.FinStronglyMeasurable (f ⊓ g) μ
-- MeasureTheory.finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite
theorem finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite {α β} {f : α → β}
    [TopologicalSpace β] [T2Space β] [Zero β] {_ : MeasurableSpace α} {μ : Measure α} :
    FinStronglyMeasurable f μ ↔
      StronglyMeasurable f ∧
        ∃ t, MeasurableSet t ∧ (∀ x ∈ tᶜ, f x = 0) ∧ SigmaFinite (μ.restrict t) 
Arguments: {α : Type u_5} {β : Type u_6} {f : α → β} [TopologicalSpace β] [T2Space β] [Zero β]
Goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α},
  MeasureTheory.FinStronglyMeasurable f μ ↔
    MeasureTheory.StronglyMeasurable f ∧
      ∃ t, MeasurableSet t ∧ (∀ x ∈ tᶜ, f x = 0) ∧ MeasureTheory.SigmaFinite (μ.restrict t)
-- MeasureTheory.aefinStronglyMeasurable_zero
theorem aefinStronglyMeasurable_zero {α β} {_ : MeasurableSpace α} (μ : Measure α) [Zero β]
    [TopologicalSpace β] : AEFinStronglyMeasurable (0 : α → β) μ 
Arguments: {α : Type u_5} {β : Type u_6}
Goal: ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α) [inst : Zero β] [inst_1 : TopologicalSpace β],
  MeasureTheory.AEFinStronglyMeasurable 0 μ
-- MeasureTheory.aestronglyMeasurable_const
@[measurability]
theorem aestronglyMeasurable_const {α β} {_ : MeasurableSpace α} {μ : Measure α}
    [TopologicalSpace β] {b : β} : AEStronglyMeasurable (fun _ : α => b) μ 
Arguments: {α : Type u_5} {β : Type u_6}
Goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : TopologicalSpace β] {b : β},
  MeasureTheory.AEStronglyMeasurable (fun x => b) μ
-- MeasureTheory.aestronglyMeasurable_zero
@[to_additive (attr := measurability)]
theorem aestronglyMeasurable_one {α β} {_ : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
    [One β] : AEStronglyMeasurable (1 : α → β) μ 
Arguments: {α : Type u_5} {β : Type u_6}
Goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : TopologicalSpace β] [inst_1 : Zero β],
  MeasureTheory.AEStronglyMeasurable 0 μ
-- MeasureTheory.aestronglyMeasurable_one
@[to_additive (attr := measurability)]
theorem aestronglyMeasurable_one {α β} {_ : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
    [One β] : AEStronglyMeasurable (1 : α → β) μ 
Arguments: {α : Type u_5} {β : Type u_6}
Goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : TopologicalSpace β] [inst_1 : One β],
  MeasureTheory.AEStronglyMeasurable 1 μ
-- MeasureTheory.Subsingleton.aestronglyMeasurable
@[simp]
theorem Subsingleton.aestronglyMeasurable {_ : MeasurableSpace α} [TopologicalSpace β]
    [Subsingleton β] {μ : Measure α} (f : α → β) : AEStronglyMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2}
Goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] [inst_1 : Subsingleton β] {μ : MeasureTheory.Measure α}
  (f : α → β), MeasureTheory.AEStronglyMeasurable f μ
-- MeasureTheory.Subsingleton.aestronglyMeasurable'
@[simp]
theorem Subsingleton.aestronglyMeasurable' {_ : MeasurableSpace α} [TopologicalSpace β]
    [Subsingleton α] {μ : Measure α} (f : α → β) : AEStronglyMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2}
Goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] [inst_1 : Subsingleton α] {μ : MeasureTheory.Measure α}
  (f : α → β), MeasureTheory.AEStronglyMeasurable f μ
-- MeasureTheory.aestronglyMeasurable_zero_measure
@[simp]
theorem aestronglyMeasurable_zero_measure [MeasurableSpace α] [TopologicalSpace β] (f : α → β) :
    AEStronglyMeasurable f (0 : Measure α) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace β] (f : α → β)
Goal: MeasureTheory.AEStronglyMeasurable f 0
-- MeasureTheory.SimpleFunc.aestronglyMeasurable
@[measurability]
theorem SimpleFunc.aestronglyMeasurable {_ : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
    (f : α →ₛ β) : AEStronglyMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2}
Goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : TopologicalSpace β] (f : MeasureTheory.SimpleFunc α β),
  MeasureTheory.AEStronglyMeasurable (↑f) μ
-- MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mk
theorem stronglyMeasurable_mk (hf : AEStronglyMeasurable f μ) : StronglyMeasurable (hf.mk f) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.StronglyMeasurable (MeasureTheory.AEStronglyMeasurable.mk f hf)
-- MeasureTheory.AEStronglyMeasurable.measurable_mk
theorem measurable_mk [PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β]
    (hf : AEStronglyMeasurable f μ) : Measurable (hf.mk f) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [TopologicalSpace.PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β] (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: Measurable (MeasureTheory.AEStronglyMeasurable.mk f hf)
-- MeasureTheory.AEStronglyMeasurable.ae_eq_mk
theorem ae_eq_mk (hf : AEStronglyMeasurable f μ) : f =ᵐ[μ] hf.mk f 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: f =ᵐ[μ] MeasureTheory.AEStronglyMeasurable.mk f hf
-- MeasureTheory.AEStronglyMeasurable.aemeasurable
@[aesop 5% apply (rule_sets := [Measurable])]
protected theorem aemeasurable {β} [MeasurableSpace β] [TopologicalSpace β]
    [PseudoMetrizableSpace β] [BorelSpace β] {f : α → β} (hf : AEStronglyMeasurable f μ) :
    AEMeasurable f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_5} [MeasurableSpace β] [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [BorelSpace β] {f : α → β} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: AEMeasurable f μ
-- MeasureTheory.AEStronglyMeasurable.congr
theorem congr (hf : AEStronglyMeasurable f μ) (h : f =ᵐ[μ] g) : AEStronglyMeasurable g μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f g : α → β} (hf : MeasureTheory.AEStronglyMeasurable f μ) (h : f =ᵐ[μ] g)
Goal: MeasureTheory.AEStronglyMeasurable g μ
-- aestronglyMeasurable_congr
theorem _root_.aestronglyMeasurable_congr (h : f =ᵐ[μ] g) :
    AEStronglyMeasurable f μ ↔ AEStronglyMeasurable g μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f g : α → β} (h : f =ᵐ[μ] g)
Goal: MeasureTheory.AEStronglyMeasurable f μ ↔ MeasureTheory.AEStronglyMeasurable g μ
-- MeasureTheory.AEStronglyMeasurable.mono_measure
theorem mono_measure {ν : Measure α} (hf : AEStronglyMeasurable f μ) (h : ν ≤ μ) :
    AEStronglyMeasurable f ν 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} {ν : MeasureTheory.Measure α} (hf : MeasureTheory.AEStronglyMeasurable f μ) (h : ν ≤ μ)
Goal: MeasureTheory.AEStronglyMeasurable f ν
-- MeasureTheory.AEStronglyMeasurable.mono_ac
protected lemma mono_ac (h : ν ≪ μ) (hμ : AEStronglyMeasurable f μ) : AEStronglyMeasurable f ν 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} (h : ν ≪ μ) (hμ : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable f ν
-- MeasureTheory.AEStronglyMeasurable.mono_set
theorem mono_set {s t} (h : s ⊆ t) (ht : AEStronglyMeasurable f (μ.restrict t)) :
    AEStronglyMeasurable f (μ.restrict s) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} {s t : Set α} (h : s ⊆ t) (ht : MeasureTheory.AEStronglyMeasurable f (μ.restrict t))
Goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict s)
-- MeasureTheory.AEStronglyMeasurable.restrict
protected theorem restrict (hfm : AEStronglyMeasurable f μ) {s} :
    AEStronglyMeasurable f (μ.restrict s) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} (hfm : MeasureTheory.AEStronglyMeasurable f μ) {s : Set α}
Goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict s)
-- MeasureTheory.AEStronglyMeasurable.ae_mem_imp_eq_mk
theorem ae_mem_imp_eq_mk {s} (h : AEStronglyMeasurable f (μ.restrict s)) :
    ∀ᵐ x ∂μ, x ∈ s → f x = h.mk f x 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} {s : Set α} (h : MeasureTheory.AEStronglyMeasurable f (μ.restrict s))
Goal: ∀ᵐ (x : α) ∂μ, x ∈ s → f x = MeasureTheory.AEStronglyMeasurable.mk f h x
-- Continuous.comp_aestronglyMeasurable
theorem _root_.Continuous.comp_aestronglyMeasurable {g : β → γ} {f : α → β} (hg : Continuous g)
    (hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable (fun x => g (f x)) μ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] {g : β → γ} {f : α → β} (hg : Continuous g) (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => g (f x)) μ
-- Continuous.aestronglyMeasurable
theorem _root_.Continuous.aestronglyMeasurable [TopologicalSpace α] [OpensMeasurableSpace α]
    [PseudoMetrizableSpace β] [SecondCountableTopologyEither α β] (hf : Continuous f) :
    AEStronglyMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [TopologicalSpace α] [OpensMeasurableSpace α] [TopologicalSpace.PseudoMetrizableSpace β] [SecondCountableTopologyEither α β] (hf : Continuous f)
Goal: MeasureTheory.AEStronglyMeasurable f μ
-- MeasureTheory.AEStronglyMeasurable.prod_mk
protected theorem prod_mk {f : α → β} {g : α → γ} (hf : AEStronglyMeasurable f μ)
    (hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (fun x => (f x, g x)) μ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] {f : α → β} {g : α → γ} (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => (f x, g x)) μ
-- Continuous.comp_aestronglyMeasurable₂
theorem _root_.Continuous.comp_aestronglyMeasurable₂
    {β' : Type*} [TopologicalSpace β']
    {g : β → β' → γ} {f : α → β} {f' : α → β'} (hg : Continuous g.uncurry)
    (hf : AEStronglyMeasurable f μ) (h'f : AEStronglyMeasurable f' μ) :
    AEStronglyMeasurable (fun x => g (f x) (f' x)) μ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] [TopologicalSpace γ] {β' : Type u_5} [TopologicalSpace β'] {g : β → β' → γ} {f : α → β} {f' : α → β'} (hg : Continuous (Function.uncurry g)) (hf : MeasureTheory.AEStronglyMeasurable f μ) (h'f : MeasureTheory.AEStronglyMeasurable f' μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => g (f x) (f' x)) μ
-- Measurable.aestronglyMeasurable
@[fun_prop, aesop unsafe 30% apply (rule_sets := [Measurable])]
theorem _root_.Measurable.aestronglyMeasurable {_ : MeasurableSpace α} {μ : Measure α}
    [MeasurableSpace β] [PseudoMetrizableSpace β] [SecondCountableTopology β]
    [OpensMeasurableSpace β] (hf : Measurable f) : AEStronglyMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace β] {f : α → β}
Goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasurableSpace β]
  [inst_1 : TopologicalSpace.PseudoMetrizableSpace β] [inst_2 : SecondCountableTopology β]
  [inst_3 : OpensMeasurableSpace β], Measurable f → MeasureTheory.AEStronglyMeasurable f μ