ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Measurable.add_stronglyMeasurable
theorem _root_.Measurable.add_stronglyMeasurable
    {α E : Type*} {_ : MeasurableSpace α} [AddGroup E] [TopologicalSpace E]
    [MeasurableSpace E] [BorelSpace E] [ContinuousAdd E] [PseudoMetrizableSpace E]
    {g f : α → E} (hg : Measurable g) (hf : StronglyMeasurable f) :
    Measurable (g + f) 
Arguments: {α : Type u_5} {E : Type u_6}
Goal: ∀ {x : MeasurableSpace α} [inst : AddGroup E] [inst_1 : TopologicalSpace E] [inst_2 : MeasurableSpace E]
  [inst_3 : BorelSpace E] [inst_4 : ContinuousAdd E] [inst_5 : TopologicalSpace.PseudoMetrizableSpace E] {g f : α → E},
  Measurable g → MeasureTheory.StronglyMeasurable f → Measurable (g + f)
-- Measurable.sub_stronglyMeasurable
theorem _root_.Measurable.sub_stronglyMeasurable
    {α E : Type*} {_ : MeasurableSpace α} [AddCommGroup E] [TopologicalSpace E]
    [MeasurableSpace E] [BorelSpace E] [ContinuousAdd E] [ContinuousNeg E] [PseudoMetrizableSpace E]
    {g f : α → E} (hg : Measurable g) (hf : StronglyMeasurable f) :
    Measurable (g - f) 
Arguments: {α : Type u_5} {E : Type u_6}
Goal: ∀ {x : MeasurableSpace α} [inst : AddCommGroup E] [inst_1 : TopologicalSpace E] [inst_2 : MeasurableSpace E]
  [inst_3 : BorelSpace E] [inst_4 : ContinuousAdd E] [inst_5 : ContinuousNeg E]
  [inst_6 : TopologicalSpace.PseudoMetrizableSpace E] {g f : α → E},
  Measurable g → MeasureTheory.StronglyMeasurable f → Measurable (g - f)
-- Measurable.stronglyMeasurable_add
theorem _root_.Measurable.stronglyMeasurable_add
    {α E : Type*} {_ : MeasurableSpace α} [AddGroup E] [TopologicalSpace E]
    [MeasurableSpace E] [BorelSpace E] [ContinuousAdd E] [PseudoMetrizableSpace E]
    {g f : α → E} (hg : Measurable g) (hf : StronglyMeasurable f) :
    Measurable (f + g) 
Arguments: {α : Type u_5} {E : Type u_6}
Goal: ∀ {x : MeasurableSpace α} [inst : AddGroup E] [inst_1 : TopologicalSpace E] [inst_2 : MeasurableSpace E]
  [inst_3 : BorelSpace E] [inst_4 : ContinuousAdd E] [inst_5 : TopologicalSpace.PseudoMetrizableSpace E] {g f : α → E},
  Measurable g → MeasureTheory.StronglyMeasurable f → Measurable (f + g)
-- stronglyMeasurable_const_smul_iff
theorem _root_.stronglyMeasurable_const_smul_iff {m : MeasurableSpace α} (c : G) :
    (StronglyMeasurable fun x => c • f x) ↔ StronglyMeasurable f 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {G : Type u_6} [TopologicalSpace β] [Group G] [MulAction G β] [ContinuousConstSMul G β] {m : MeasurableSpace α} (c : G)
Goal: (MeasureTheory.StronglyMeasurable fun x => c • f x) ↔ MeasureTheory.StronglyMeasurable f
-- IsUnit.stronglyMeasurable_const_smul_iff
nonrec theorem _root_.IsUnit.stronglyMeasurable_const_smul_iff {_ : MeasurableSpace α} {c : M}
    (hc : IsUnit c) :
    (StronglyMeasurable fun x => c • f x) ↔ StronglyMeasurable f 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {M : Type u_5} [TopologicalSpace β] [Monoid M] [MulAction M β] [ContinuousConstSMul M β]
Goal: ∀ {x : MeasurableSpace α} {c : M},
  IsUnit c → ((MeasureTheory.StronglyMeasurable fun x => c • f x) ↔ MeasureTheory.StronglyMeasurable f)
-- stronglyMeasurable_const_smul_iff₀
theorem _root_.stronglyMeasurable_const_smul_iff₀ {_ : MeasurableSpace α} {c : G₀} (hc : c ≠ 0) :
    (StronglyMeasurable fun x => c • f x) ↔ StronglyMeasurable f 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {G₀ : Type u_7} [TopologicalSpace β] [GroupWithZero G₀] [MulAction G₀ β] [ContinuousConstSMul G₀ β]
Goal: ∀ {x : MeasurableSpace α} {c : G₀},
  c ≠ 0 → ((MeasureTheory.StronglyMeasurable fun x => c • f x) ↔ MeasureTheory.StronglyMeasurable f)
-- MeasureTheory.StronglyMeasurable.sup
@[aesop safe 20 (rule_sets := [Measurable])]
protected theorem sup [Sup β] [ContinuousSup β] (hf : StronglyMeasurable f)
    (hg : StronglyMeasurable g) : StronglyMeasurable (f ⊔ g) 
Arguments: {α : Type u_1} {β : Type u_2} {f g : α → β} [MeasurableSpace α] [TopologicalSpace β] [Sup β] [ContinuousSup β] (hf : MeasureTheory.StronglyMeasurable f) (hg : MeasureTheory.StronglyMeasurable g)
Goal: MeasureTheory.StronglyMeasurable (f ⊔ g)
-- MeasureTheory.StronglyMeasurable.inf
@[aesop safe 20 (rule_sets := [Measurable])]
protected theorem inf [Inf β] [ContinuousInf β] (hf : StronglyMeasurable f)
    (hg : StronglyMeasurable g) : StronglyMeasurable (f ⊓ g) 
Arguments: {α : Type u_1} {β : Type u_2} {f g : α → β} [MeasurableSpace α] [TopologicalSpace β] [Inf β] [ContinuousInf β] (hf : MeasureTheory.StronglyMeasurable f) (hg : MeasureTheory.StronglyMeasurable g)
Goal: MeasureTheory.StronglyMeasurable (f ⊓ g)
-- List.stronglyMeasurable_sum'
@[to_additive (attr := measurability)]
theorem _root_.List.stronglyMeasurable_prod' (l : List (α → M))
    (hl : ∀ f ∈ l, StronglyMeasurable f) : StronglyMeasurable l.prod 
Arguments: {α : Type u_1} {M : Type u_5} [AddMonoid M] [TopologicalSpace M] [ContinuousAdd M] {m : MeasurableSpace α} (l : List (α → M)) (hl : ∀ f ∈ l, MeasureTheory.StronglyMeasurable f)
Goal: MeasureTheory.StronglyMeasurable l.sum
-- List.stronglyMeasurable_prod'
@[to_additive (attr := measurability)]
theorem _root_.List.stronglyMeasurable_prod' (l : List (α → M))
    (hl : ∀ f ∈ l, StronglyMeasurable f) : StronglyMeasurable l.prod 
Arguments: {α : Type u_1} {M : Type u_5} [Monoid M] [TopologicalSpace M] [ContinuousMul M] {m : MeasurableSpace α} (l : List (α → M)) (hl : ∀ f ∈ l, MeasureTheory.StronglyMeasurable f)
Goal: MeasureTheory.StronglyMeasurable l.prod
-- List.stronglyMeasurable_prod
@[to_additive (attr := measurability)]
theorem _root_.List.stronglyMeasurable_prod (l : List (α → M))
    (hl : ∀ f ∈ l, StronglyMeasurable f) :
    StronglyMeasurable fun x => (l.map fun f : α → M => f x).prod 
Arguments: {α : Type u_1} {M : Type u_5} [Monoid M] [TopologicalSpace M] [ContinuousMul M] {m : MeasurableSpace α} (l : List (α → M)) (hl : ∀ f ∈ l, MeasureTheory.StronglyMeasurable f)
Goal: MeasureTheory.StronglyMeasurable fun x => (List.map (fun f => f x) l).prod
-- List.stronglyMeasurable_sum
@[to_additive (attr := measurability)]
theorem _root_.List.stronglyMeasurable_prod (l : List (α → M))
    (hl : ∀ f ∈ l, StronglyMeasurable f) :
    StronglyMeasurable fun x => (l.map fun f : α → M => f x).prod 
Arguments: {α : Type u_1} {M : Type u_5} [AddMonoid M] [TopologicalSpace M] [ContinuousAdd M] {m : MeasurableSpace α} (l : List (α → M)) (hl : ∀ f ∈ l, MeasureTheory.StronglyMeasurable f)
Goal: MeasureTheory.StronglyMeasurable fun x => (List.map (fun f => f x) l).sum
-- Multiset.stronglyMeasurable_prod'
@[to_additive (attr := measurability)]
theorem _root_.Multiset.stronglyMeasurable_prod' (l : Multiset (α → M))
    (hl : ∀ f ∈ l, StronglyMeasurable f) : StronglyMeasurable l.prod 
Arguments: {α : Type u_1} {M : Type u_5} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M] {m : MeasurableSpace α} (l : Multiset (α → M)) (hl : ∀ f ∈ l, MeasureTheory.StronglyMeasurable f)
Goal: MeasureTheory.StronglyMeasurable l.prod
-- Multiset.stronglyMeasurable_sum'
@[to_additive (attr := measurability)]
theorem _root_.Multiset.stronglyMeasurable_prod' (l : Multiset (α → M))
    (hl : ∀ f ∈ l, StronglyMeasurable f) : StronglyMeasurable l.prod 
Arguments: {α : Type u_1} {M : Type u_5} [AddCommMonoid M] [TopologicalSpace M] [ContinuousAdd M] {m : MeasurableSpace α} (l : Multiset (α → M)) (hl : ∀ f ∈ l, MeasureTheory.StronglyMeasurable f)
Goal: MeasureTheory.StronglyMeasurable l.sum
-- Multiset.stronglyMeasurable_sum
@[to_additive (attr := measurability)]
theorem _root_.Multiset.stronglyMeasurable_prod (s : Multiset (α → M))
    (hs : ∀ f ∈ s, StronglyMeasurable f) :
    StronglyMeasurable fun x => (s.map fun f : α → M => f x).prod 
Arguments: {α : Type u_1} {M : Type u_5} [AddCommMonoid M] [TopologicalSpace M] [ContinuousAdd M] {m : MeasurableSpace α} (s : Multiset (α → M)) (hs : ∀ f ∈ s, MeasureTheory.StronglyMeasurable f)
Goal: MeasureTheory.StronglyMeasurable fun x => (Multiset.map (fun f => f x) s).sum
-- Multiset.stronglyMeasurable_prod
@[to_additive (attr := measurability)]
theorem _root_.Multiset.stronglyMeasurable_prod (s : Multiset (α → M))
    (hs : ∀ f ∈ s, StronglyMeasurable f) :
    StronglyMeasurable fun x => (s.map fun f : α → M => f x).prod 
Arguments: {α : Type u_1} {M : Type u_5} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M] {m : MeasurableSpace α} (s : Multiset (α → M)) (hs : ∀ f ∈ s, MeasureTheory.StronglyMeasurable f)
Goal: MeasureTheory.StronglyMeasurable fun x => (Multiset.map (fun f => f x) s).prod
-- Finset.stronglyMeasurable_sum'
@[to_additive (attr := measurability)]
theorem _root_.Finset.stronglyMeasurable_prod' {ι : Type*} {f : ι → α → M} (s : Finset ι)
    (hf : ∀ i ∈ s, StronglyMeasurable (f i)) : StronglyMeasurable (∏ i ∈ s, f i) 
Arguments: {α : Type u_1} {M : Type u_5} [AddCommMonoid M] [TopologicalSpace M] [ContinuousAdd M] {m : MeasurableSpace α} {ι : Type u_6} {f : ι → α → M} (s : Finset ι) (hf : ∀ i ∈ s, MeasureTheory.StronglyMeasurable (f i))
Goal: MeasureTheory.StronglyMeasurable (∑ i ∈ s, f i)
-- Finset.stronglyMeasurable_prod'
@[to_additive (attr := measurability)]
theorem _root_.Finset.stronglyMeasurable_prod' {ι : Type*} {f : ι → α → M} (s : Finset ι)
    (hf : ∀ i ∈ s, StronglyMeasurable (f i)) : StronglyMeasurable (∏ i ∈ s, f i) 
Arguments: {α : Type u_1} {M : Type u_5} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M] {m : MeasurableSpace α} {ι : Type u_6} {f : ι → α → M} (s : Finset ι) (hf : ∀ i ∈ s, MeasureTheory.StronglyMeasurable (f i))
Goal: MeasureTheory.StronglyMeasurable (∏ i ∈ s, f i)
-- Finset.stronglyMeasurable_sum
@[to_additive (attr := measurability)]
theorem _root_.Finset.stronglyMeasurable_prod {ι : Type*} {f : ι → α → M} (s : Finset ι)
    (hf : ∀ i ∈ s, StronglyMeasurable (f i)) : StronglyMeasurable fun a => ∏ i ∈ s, f i a 
Arguments: {α : Type u_1} {M : Type u_5} [AddCommMonoid M] [TopologicalSpace M] [ContinuousAdd M] {m : MeasurableSpace α} {ι : Type u_6} {f : ι → α → M} (s : Finset ι) (hf : ∀ i ∈ s, MeasureTheory.StronglyMeasurable (f i))
Goal: MeasureTheory.StronglyMeasurable fun a => ∑ i ∈ s, f i a
-- Finset.stronglyMeasurable_prod
@[to_additive (attr := measurability)]
theorem _root_.Finset.stronglyMeasurable_prod {ι : Type*} {f : ι → α → M} (s : Finset ι)
    (hf : ∀ i ∈ s, StronglyMeasurable (f i)) : StronglyMeasurable fun a => ∏ i ∈ s, f i a 
Arguments: {α : Type u_1} {M : Type u_5} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M] {m : MeasurableSpace α} {ι : Type u_6} {f : ι → α → M} (s : Finset ι) (hf : ∀ i ∈ s, MeasureTheory.StronglyMeasurable (f i))
Goal: MeasureTheory.StronglyMeasurable fun a => ∏ i ∈ s, f i a
-- MeasureTheory.StronglyMeasurable.isSeparable_range
protected theorem isSeparable_range {m : MeasurableSpace α} [TopologicalSpace β]
    (hf : StronglyMeasurable f) : TopologicalSpace.IsSeparable (range f) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {m : MeasurableSpace α} [TopologicalSpace β] (hf : MeasureTheory.StronglyMeasurable f)
Goal: TopologicalSpace.IsSeparable (Set.range f)
-- MeasureTheory.StronglyMeasurable.separableSpace_range_union_singleton
theorem separableSpace_range_union_singleton {_ : MeasurableSpace α} [TopologicalSpace β]
    [PseudoMetrizableSpace β] (hf : StronglyMeasurable f) {b : β} :
    SeparableSpace (range f ∪ {b} : Set β) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β}
Goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] [inst_1 : TopologicalSpace.PseudoMetrizableSpace β],
  MeasureTheory.StronglyMeasurable f → ∀ {b : β}, TopologicalSpace.SeparableSpace ↑(Set.range f ∪ {b})
-- Measurable.stronglyMeasurable
@[aesop 90% apply (rule_sets := [Measurable])]
theorem _root_.Measurable.stronglyMeasurable [TopologicalSpace β] [PseudoMetrizableSpace β]
    [SecondCountableTopology β] [OpensMeasurableSpace β] (hf : Measurable f) :
    StronglyMeasurable f 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {mα : MeasurableSpace α} [MeasurableSpace β] [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [SecondCountableTopology β] [OpensMeasurableSpace β] (hf : Measurable f)
Goal: MeasureTheory.StronglyMeasurable f
-- stronglyMeasurable_iff_measurable
theorem _root_.stronglyMeasurable_iff_measurable [TopologicalSpace β] [MetrizableSpace β]
    [BorelSpace β] [SecondCountableTopology β] : StronglyMeasurable f ↔ Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {mα : MeasurableSpace α} [MeasurableSpace β] [TopologicalSpace β] [TopologicalSpace.MetrizableSpace β] [BorelSpace β] [SecondCountableTopology β]
Goal: MeasureTheory.StronglyMeasurable f ↔ Measurable f
-- stronglyMeasurable_id
@[measurability]
theorem _root_.stronglyMeasurable_id [TopologicalSpace α] [PseudoMetrizableSpace α]
    [OpensMeasurableSpace α] [SecondCountableTopology α] : StronglyMeasurable (id : α → α) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} [TopologicalSpace α] [TopologicalSpace.PseudoMetrizableSpace α] [OpensMeasurableSpace α] [SecondCountableTopology α]
Goal: MeasureTheory.StronglyMeasurable id
-- stronglyMeasurable_iff_measurable_separable
theorem _root_.stronglyMeasurable_iff_measurable_separable {m : MeasurableSpace α}
    [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β] :
    StronglyMeasurable f ↔ Measurable f ∧ IsSeparable (range f) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {m : MeasurableSpace α} [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β]
Goal: MeasureTheory.StronglyMeasurable f ↔ Measurable f ∧ TopologicalSpace.IsSeparable (Set.range f)
-- Continuous.stronglyMeasurable
theorem _root_.Continuous.stronglyMeasurable [MeasurableSpace α] [TopologicalSpace α]
    [OpensMeasurableSpace α] [TopologicalSpace β] [PseudoMetrizableSpace β]
    [h : SecondCountableTopologyEither α β] {f : α → β} (hf : Continuous f) :
    StronglyMeasurable f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [h : SecondCountableTopologyEither α β] {f : α → β} (hf : Continuous f)
Goal: MeasureTheory.StronglyMeasurable f
-- Continuous.stronglyMeasurable_of_support_subset_isCompact
@[to_additive]
theorem _root_.Continuous.stronglyMeasurable_of_mulSupport_subset_isCompact
    [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] [MeasurableSpace β]
    [TopologicalSpace β] [PseudoMetrizableSpace β] [BorelSpace β] [One β] {f : α → β}
    (hf : Continuous f) {k : Set α} (hk : IsCompact k)
    (h'f : mulSupport f ⊆ k) : StronglyMeasurable f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [BorelSpace β] [Zero β] {f : α → β} (hf : Continuous f) {k : Set α} (hk : IsCompact k) (h'f : Function.support f ⊆ k)
Goal: MeasureTheory.StronglyMeasurable f
-- Continuous.stronglyMeasurable_of_mulSupport_subset_isCompact
@[to_additive]
theorem _root_.Continuous.stronglyMeasurable_of_mulSupport_subset_isCompact
    [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] [MeasurableSpace β]
    [TopologicalSpace β] [PseudoMetrizableSpace β] [BorelSpace β] [One β] {f : α → β}
    (hf : Continuous f) {k : Set α} (hk : IsCompact k)
    (h'f : mulSupport f ⊆ k) : StronglyMeasurable f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [BorelSpace β] [One β] {f : α → β} (hf : Continuous f) {k : Set α} (hk : IsCompact k) (h'f : Function.mulSupport f ⊆ k)
Goal: MeasureTheory.StronglyMeasurable f
-- Continuous.stronglyMeasurable_of_hasCompactMulSupport
@[to_additive]
theorem _root_.Continuous.stronglyMeasurable_of_hasCompactMulSupport
    [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] [MeasurableSpace β]
    [TopologicalSpace β] [PseudoMetrizableSpace β] [BorelSpace β] [One β] {f : α → β}
    (hf : Continuous f) (h'f : HasCompactMulSupport f) : StronglyMeasurable f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [BorelSpace β] [One β] {f : α → β} (hf : Continuous f) (h'f : HasCompactMulSupport f)
Goal: MeasureTheory.StronglyMeasurable f
-- Continuous.stronglyMeasurable_of_hasCompactSupport
@[to_additive]
theorem _root_.Continuous.stronglyMeasurable_of_hasCompactMulSupport
    [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] [MeasurableSpace β]
    [TopologicalSpace β] [PseudoMetrizableSpace β] [BorelSpace β] [One β] {f : α → β}
    (hf : Continuous f) (h'f : HasCompactMulSupport f) : StronglyMeasurable f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [BorelSpace β] [Zero β] {f : α → β} (hf : Continuous f) (h'f : HasCompactSupport f)
Goal: MeasureTheory.StronglyMeasurable f
-- HasCompactSupport.stronglyMeasurable_of_prod
lemma _root_.HasCompactSupport.stronglyMeasurable_of_prod {X Y : Type*} [Zero α]
    [TopologicalSpace X] [TopologicalSpace Y] [MeasurableSpace X] [MeasurableSpace Y]
    [OpensMeasurableSpace X] [OpensMeasurableSpace Y] [TopologicalSpace α] [PseudoMetrizableSpace α]
    {f : X × Y → α} (hf : Continuous f) (h'f : HasCompactSupport f) :
    StronglyMeasurable f 
Arguments: {α : Type u_1} {X : Type u_5} {Y : Type u_6} [Zero α] [TopologicalSpace X] [TopologicalSpace Y] [MeasurableSpace X] [MeasurableSpace Y] [OpensMeasurableSpace X] [OpensMeasurableSpace Y] [TopologicalSpace α] [TopologicalSpace.PseudoMetrizableSpace α] {f : X × Y → α} (hf : Continuous f) (h'f : HasCompactSupport f)
Goal: MeasureTheory.StronglyMeasurable f
-- Embedding.comp_stronglyMeasurable_iff
theorem _root_.Embedding.comp_stronglyMeasurable_iff {m : MeasurableSpace α} [TopologicalSpace β]
    [PseudoMetrizableSpace β] [TopologicalSpace γ] [PseudoMetrizableSpace γ] {g : β → γ} {f : α → β}
    (hg : Embedding g) : (StronglyMeasurable fun x => g (f x)) ↔ StronglyMeasurable f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : MeasurableSpace α} [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [TopologicalSpace γ] [TopologicalSpace.PseudoMetrizableSpace γ] {g : β → γ} {f : α → β} (hg : Embedding g)
Goal: (MeasureTheory.StronglyMeasurable fun x => g (f x)) ↔ MeasureTheory.StronglyMeasurable f
-- stronglyMeasurable_of_tendsto
theorem _root_.stronglyMeasurable_of_tendsto {ι : Type*} {m : MeasurableSpace α}
    [TopologicalSpace β] [PseudoMetrizableSpace β] (u : Filter ι) [NeBot u] [IsCountablyGenerated u]
    {f : ι → α → β} {g : α → β} (hf : ∀ i, StronglyMeasurable (f i)) (lim : Tendsto f u (𝓝 g)) :
    StronglyMeasurable g 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Type u_5} {m : MeasurableSpace α} [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] (u : Filter ι) [u.NeBot] [u.IsCountablyGenerated] {f : ι → α → β} {g : α → β} (hf : ∀ (i : ι), MeasureTheory.StronglyMeasurable (f i)) (lim : Filter.Tendsto f u (𝓝 g))
Goal: MeasureTheory.StronglyMeasurable g
-- MeasureTheory.StronglyMeasurable.piecewise
protected theorem piecewise {m : MeasurableSpace α} [TopologicalSpace β] {s : Set α}
    {_ : DecidablePred (· ∈ s)} (hs : MeasurableSet s) (hf : StronglyMeasurable f)
    (hg : StronglyMeasurable g) : StronglyMeasurable (Set.piecewise s f g) 
Arguments: {α : Type u_1} {β : Type u_2} {f g : α → β} {m : MeasurableSpace α} [TopologicalSpace β] {s : Set α}
Goal: ∀ {x : DecidablePred fun x => x ∈ s},
  MeasurableSet s →
    MeasureTheory.StronglyMeasurable f →
      MeasureTheory.StronglyMeasurable g → MeasureTheory.StronglyMeasurable (s.piecewise f g)
-- MeasureTheory.StronglyMeasurable.ite
protected theorem ite {_ : MeasurableSpace α} [TopologicalSpace β] {p : α → Prop}
    {_ : DecidablePred p} (hp : MeasurableSet { a : α | p a }) (hf : StronglyMeasurable f)
    (hg : StronglyMeasurable g) : StronglyMeasurable fun x => ite (p x) (f x) (g x) 
Arguments: {α : Type u_1} {β : Type u_2} {f g : α → β}
Goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] {p : α → Prop} {x_1 : DecidablePred p},
  MeasurableSet {a | p a} →
    MeasureTheory.StronglyMeasurable f →
      MeasureTheory.StronglyMeasurable g → MeasureTheory.StronglyMeasurable fun x => if p x then f x else g x
-- MeasurableEmbedding.stronglyMeasurable_extend
@[measurability]
theorem _root_.MeasurableEmbedding.stronglyMeasurable_extend {f : α → β} {g : α → γ} {g' : γ → β}
    {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [TopologicalSpace β]
    (hg : MeasurableEmbedding g) (hf : StronglyMeasurable f) (hg' : StronglyMeasurable g') :
    StronglyMeasurable (Function.extend g f g') 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β} {g : α → γ} {g' : γ → β} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [TopologicalSpace β] (hg : MeasurableEmbedding g) (hf : MeasureTheory.StronglyMeasurable f) (hg' : MeasureTheory.StronglyMeasurable g')
Goal: MeasureTheory.StronglyMeasurable (Function.extend g f g')
-- MeasurableEmbedding.exists_stronglyMeasurable_extend
theorem _root_.MeasurableEmbedding.exists_stronglyMeasurable_extend {f : α → β} {g : α → γ}
    {_ : MeasurableSpace α} {_ : MeasurableSpace γ} [TopologicalSpace β]
    (hg : MeasurableEmbedding g) (hf : StronglyMeasurable f) (hne : γ → Nonempty β) :
    ∃ f' : γ → β, StronglyMeasurable f' ∧ f' ∘ g = f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β} {g : α → γ}
Goal: ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace γ} [inst : TopologicalSpace β],
  MeasurableEmbedding g →
    MeasureTheory.StronglyMeasurable f → (γ → Nonempty β) → ∃ f', MeasureTheory.StronglyMeasurable f' ∧ f' ∘ g = f
-- stronglyMeasurable_of_stronglyMeasurable_union_cover
theorem _root_.stronglyMeasurable_of_stronglyMeasurable_union_cover {m : MeasurableSpace α}
    [TopologicalSpace β] {f : α → β} (s t : Set α) (hs : MeasurableSet s) (ht : MeasurableSet t)
    (h : univ ⊆ s ∪ t) (hc : StronglyMeasurable fun a : s => f a)
    (hd : StronglyMeasurable fun a : t => f a) : StronglyMeasurable f 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [TopologicalSpace β] {f : α → β} (s t : Set α) (hs : MeasurableSet s) (ht : MeasurableSet t) (h : Set.univ ⊆ s ∪ t) (hc : MeasureTheory.StronglyMeasurable fun a => f ↑a) (hd : MeasureTheory.StronglyMeasurable fun a => f ↑a)
Goal: MeasureTheory.StronglyMeasurable f
-- stronglyMeasurable_of_restrict_of_restrict_compl
theorem _root_.stronglyMeasurable_of_restrict_of_restrict_compl {_ : MeasurableSpace α}
    [TopologicalSpace β] {f : α → β} {s : Set α} (hs : MeasurableSet s)
    (h₁ : StronglyMeasurable (s.restrict f)) (h₂ : StronglyMeasurable (sᶜ.restrict f)) :
    StronglyMeasurable f 
Arguments: {α : Type u_1} {β : Type u_2}
Goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] {f : α → β} {s : Set α},
  MeasurableSet s →
    MeasureTheory.StronglyMeasurable (s.restrict f) →
      MeasureTheory.StronglyMeasurable (sᶜ.restrict f) → MeasureTheory.StronglyMeasurable f
-- MeasureTheory.StronglyMeasurable.indicator
@[measurability]
protected theorem indicator {_ : MeasurableSpace α} [TopologicalSpace β] [Zero β]
    (hf : StronglyMeasurable f) {s : Set α} (hs : MeasurableSet s) :
    StronglyMeasurable (s.indicator f) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β}
Goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] [inst_1 : Zero β],
  MeasureTheory.StronglyMeasurable f → ∀ {s : Set α}, MeasurableSet s → MeasureTheory.StronglyMeasurable (s.indicator f)
-- MeasureTheory.StronglyMeasurable.dist
@[aesop safe 20 apply (rule_sets := [Measurable])]
protected theorem dist {_ : MeasurableSpace α} {β : Type*} [PseudoMetricSpace β] {f g : α → β}
    (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) :
    StronglyMeasurable fun x => dist (f x) (g x) 
Arguments: {α : Type u_1}
Goal: ∀ {x : MeasurableSpace α} {β : Type u_5} [inst : PseudoMetricSpace β] {f g : α → β},
  MeasureTheory.StronglyMeasurable f →
    MeasureTheory.StronglyMeasurable g → MeasureTheory.StronglyMeasurable fun x => dist (f x) (g x)