ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.AEStronglyMeasurable.mul
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem mul [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ)
    (hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f * g) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f g : α → β} [Mul β] [ContinuousMul β] (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.AEStronglyMeasurable (f * g) μ
-- MeasureTheory.AEStronglyMeasurable.add
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem mul [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ)
    (hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f * g) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f g : α → β} [Add β] [ContinuousAdd β] (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.AEStronglyMeasurable (f + g) μ
-- MeasureTheory.AEStronglyMeasurable.add_const
@[to_additive (attr := measurability)]
protected theorem mul_const [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ) (c : β) :
    AEStronglyMeasurable (fun x => f x * c) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [Add β] [ContinuousAdd β] (hf : MeasureTheory.AEStronglyMeasurable f μ) (c : β)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => f x + c) μ
-- MeasureTheory.AEStronglyMeasurable.mul_const
@[to_additive (attr := measurability)]
protected theorem mul_const [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ) (c : β) :
    AEStronglyMeasurable (fun x => f x * c) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [Mul β] [ContinuousMul β] (hf : MeasureTheory.AEStronglyMeasurable f μ) (c : β)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => f x * c) μ
-- MeasureTheory.AEStronglyMeasurable.const_add
@[to_additive (attr := measurability)]
protected theorem const_mul [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ) (c : β) :
    AEStronglyMeasurable (fun x => c * f x) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [Add β] [ContinuousAdd β] (hf : MeasureTheory.AEStronglyMeasurable f μ) (c : β)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => c + f x) μ
-- MeasureTheory.AEStronglyMeasurable.const_mul
@[to_additive (attr := measurability)]
protected theorem const_mul [Mul β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ) (c : β) :
    AEStronglyMeasurable (fun x => c * f x) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [Mul β] [ContinuousMul β] (hf : MeasureTheory.AEStronglyMeasurable f μ) (c : β)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => c * f x) μ
-- MeasureTheory.AEStronglyMeasurable.inv
@[to_additive (attr := measurability)]
protected theorem inv [Inv β] [ContinuousInv β] (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable f⁻¹ μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [Inv β] [ContinuousInv β] (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable f⁻¹ μ
-- MeasureTheory.AEStronglyMeasurable.neg
@[to_additive (attr := measurability)]
protected theorem inv [Inv β] [ContinuousInv β] (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable f⁻¹ μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [Neg β] [ContinuousNeg β] (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable (-f) μ
-- MeasureTheory.AEStronglyMeasurable.sub
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem div [Group β] [TopologicalGroup β] (hf : AEStronglyMeasurable f μ)
    (hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f / g) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f g : α → β} [AddGroup β] [TopologicalAddGroup β] (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.AEStronglyMeasurable (f - g) μ
-- MeasureTheory.AEStronglyMeasurable.div
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem div [Group β] [TopologicalGroup β] (hf : AEStronglyMeasurable f μ)
    (hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f / g) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f g : α → β} [Group β] [TopologicalGroup β] (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.AEStronglyMeasurable (f / g) μ
-- MeasureTheory.AEStronglyMeasurable.mul_iff_right
@[to_additive]
theorem mul_iff_right [CommGroup β] [TopologicalGroup β] (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (f * g) 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} [CommGroup β] [TopologicalGroup β] (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable (f * g) μ ↔ MeasureTheory.AEStronglyMeasurable g μ
-- MeasureTheory.AEStronglyMeasurable.add_iff_right
@[to_additive]
theorem mul_iff_right [CommGroup β] [TopologicalGroup β] (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (f * g) 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} [AddCommGroup β] [TopologicalAddGroup β] (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable (f + g) μ ↔ MeasureTheory.AEStronglyMeasurable g μ
-- MeasureTheory.AEStronglyMeasurable.mul_iff_left
@[to_additive]
theorem mul_iff_left [CommGroup β] [TopologicalGroup β] (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (g * 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} [CommGroup β] [TopologicalGroup β] (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable (g * f) μ ↔ MeasureTheory.AEStronglyMeasurable g μ
-- MeasureTheory.AEStronglyMeasurable.add_iff_left
@[to_additive]
theorem mul_iff_left [CommGroup β] [TopologicalGroup β] (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (g * 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} [AddCommGroup β] [TopologicalAddGroup β] (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable (g + f) μ ↔ MeasureTheory.AEStronglyMeasurable g μ
-- MeasureTheory.AEStronglyMeasurable.smul
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem smul {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
    {g : α → β} (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} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {𝕜 : Type u_5} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜} {g : α → β} (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => f x • g x) μ
-- MeasureTheory.AEStronglyMeasurable.vadd
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
protected theorem smul {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
    {g : α → β} (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} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {𝕜 : Type u_5} [TopologicalSpace 𝕜] [VAdd 𝕜 β] [ContinuousVAdd 𝕜 β] {f : α → 𝕜} {g : α → β} (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => f x +ᵥ g x) μ
-- MeasureTheory.AEStronglyMeasurable.pow
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable])) const_nsmul]
protected theorem pow [Monoid β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ) (n : ℕ) :
    AEStronglyMeasurable (f ^ n) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [Monoid β] [ContinuousMul β] (hf : MeasureTheory.AEStronglyMeasurable f μ) (n : ℕ)
Goal: MeasureTheory.AEStronglyMeasurable (f ^ n) μ
-- MeasureTheory.AEStronglyMeasurable.const_nsmul
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable])) const_nsmul]
protected theorem pow [Monoid β] [ContinuousMul β] (hf : AEStronglyMeasurable f μ) (n : ℕ) :
    AEStronglyMeasurable (f ^ n) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [AddMonoid β] [ContinuousAdd β] (hf : MeasureTheory.AEStronglyMeasurable f μ) (n : ℕ)
Goal: MeasureTheory.AEStronglyMeasurable (n • f) μ
-- MeasureTheory.AEStronglyMeasurable.const_vadd
@[to_additive (attr := measurability)]
protected theorem const_smul {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β]
    (hf : AEStronglyMeasurable f μ) (c : 𝕜) : AEStronglyMeasurable (c • f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} {𝕜 : Type u_5} [VAdd 𝕜 β] [ContinuousConstVAdd 𝕜 β] (hf : MeasureTheory.AEStronglyMeasurable f μ) (c : 𝕜)
Goal: MeasureTheory.AEStronglyMeasurable (c +ᵥ f) μ
-- MeasureTheory.AEStronglyMeasurable.const_smul
@[to_additive (attr := measurability)]
protected theorem const_smul {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β]
    (hf : AEStronglyMeasurable f μ) (c : 𝕜) : AEStronglyMeasurable (c • f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} {𝕜 : Type u_5} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (hf : MeasureTheory.AEStronglyMeasurable f μ) (c : 𝕜)
Goal: MeasureTheory.AEStronglyMeasurable (c • f) μ
-- MeasureTheory.AEStronglyMeasurable.const_vadd'
@[to_additive (attr := measurability)]
protected theorem const_smul' {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β]
    (hf : AEStronglyMeasurable f μ) (c : 𝕜) : AEStronglyMeasurable (fun x => c • f x) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} {𝕜 : Type u_5} [VAdd 𝕜 β] [ContinuousConstVAdd 𝕜 β] (hf : MeasureTheory.AEStronglyMeasurable f μ) (c : 𝕜)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => c +ᵥ f x) μ
-- MeasureTheory.AEStronglyMeasurable.const_smul'
@[to_additive (attr := measurability)]
protected theorem const_smul' {𝕜} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β]
    (hf : AEStronglyMeasurable f μ) (c : 𝕜) : AEStronglyMeasurable (fun x => c • f x) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} {𝕜 : Type u_5} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (hf : MeasureTheory.AEStronglyMeasurable f μ) (c : 𝕜)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => c • f x) μ
-- MeasureTheory.AEStronglyMeasurable.vadd_const
@[to_additive (attr := measurability)]
protected theorem smul_const {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
    (hf : AEStronglyMeasurable f μ) (c : β) : AEStronglyMeasurable (fun x => f x • c) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {𝕜 : Type u_5} [TopologicalSpace 𝕜] [VAdd 𝕜 β] [ContinuousVAdd 𝕜 β] {f : α → 𝕜} (hf : MeasureTheory.AEStronglyMeasurable f μ) (c : β)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => f x +ᵥ c) μ
-- MeasureTheory.AEStronglyMeasurable.smul_const
@[to_additive (attr := measurability)]
protected theorem smul_const {𝕜} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜}
    (hf : AEStronglyMeasurable f μ) (c : β) : AEStronglyMeasurable (fun x => f x • c) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {𝕜 : Type u_5} [TopologicalSpace 𝕜] [SMul 𝕜 β] [ContinuousSMul 𝕜 β] {f : α → 𝕜} (hf : MeasureTheory.AEStronglyMeasurable f μ) (c : β)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => f x • c) μ
-- MeasureTheory.AEStronglyMeasurable.sup
@[aesop safe 20 apply (rule_sets := [Measurable])]
protected theorem sup [SemilatticeSup β] [ContinuousSup β] (hf : AEStronglyMeasurable f μ)
    (hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f ⊔ g) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f g : α → β} [SemilatticeSup β] [ContinuousSup β] (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.AEStronglyMeasurable (f ⊔ g) μ
-- MeasureTheory.AEStronglyMeasurable.inf
@[aesop safe 20 apply (rule_sets := [Measurable])]
protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : AEStronglyMeasurable f μ)
    (hg : AEStronglyMeasurable g μ) : AEStronglyMeasurable (f ⊓ g) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f g : α → β} [SemilatticeInf β] [ContinuousInf β] (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.AEStronglyMeasurable (f ⊓ g) μ
-- List.aestronglyMeasurable_sum'
@[to_additive (attr := measurability)]
theorem _root_.List.aestronglyMeasurable_prod' (l : List (α → M))
    (hl : ∀ f ∈ l, AEStronglyMeasurable f μ) : AEStronglyMeasurable l.prod μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {M : Type u_5} [AddMonoid M] [TopologicalSpace M] [ContinuousAdd M] (l : List (α → M)) (hl : ∀ f ∈ l, MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable l.sum μ
-- List.aestronglyMeasurable_prod'
@[to_additive (attr := measurability)]
theorem _root_.List.aestronglyMeasurable_prod' (l : List (α → M))
    (hl : ∀ f ∈ l, AEStronglyMeasurable f μ) : AEStronglyMeasurable l.prod μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {M : Type u_5} [Monoid M] [TopologicalSpace M] [ContinuousMul M] (l : List (α → M)) (hl : ∀ f ∈ l, MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable l.prod μ
-- List.aestronglyMeasurable_sum
@[to_additive (attr := measurability)]
theorem _root_.List.aestronglyMeasurable_prod
    (l : List (α → M)) (hl : ∀ f ∈ l, AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (fun x => (l.map fun f : α → M => f x).prod) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {M : Type u_5} [AddMonoid M] [TopologicalSpace M] [ContinuousAdd M] (l : List (α → M)) (hl : ∀ f ∈ l, MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => (List.map (fun f => f x) l).sum) μ
-- List.aestronglyMeasurable_prod
@[to_additive (attr := measurability)]
theorem _root_.List.aestronglyMeasurable_prod
    (l : List (α → M)) (hl : ∀ f ∈ l, AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (fun x => (l.map fun f : α → M => f x).prod) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {M : Type u_5} [Monoid M] [TopologicalSpace M] [ContinuousMul M] (l : List (α → M)) (hl : ∀ f ∈ l, MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => (List.map (fun f => f x) l).prod) μ
-- Multiset.aestronglyMeasurable_prod'
@[to_additive (attr := measurability)]
theorem _root_.Multiset.aestronglyMeasurable_prod' (l : Multiset (α → M))
    (hl : ∀ f ∈ l, AEStronglyMeasurable f μ) : AEStronglyMeasurable l.prod μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {M : Type u_5} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M] (l : Multiset (α → M)) (hl : ∀ f ∈ l, MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable l.prod μ
-- Multiset.aestronglyMeasurable_sum'
@[to_additive (attr := measurability)]
theorem _root_.Multiset.aestronglyMeasurable_prod' (l : Multiset (α → M))
    (hl : ∀ f ∈ l, AEStronglyMeasurable f μ) : AEStronglyMeasurable l.prod μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {M : Type u_5} [AddCommMonoid M] [TopologicalSpace M] [ContinuousAdd M] (l : Multiset (α → M)) (hl : ∀ f ∈ l, MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable l.sum μ
-- Multiset.aestronglyMeasurable_prod
@[to_additive (attr := measurability)]
theorem _root_.Multiset.aestronglyMeasurable_prod (s : Multiset (α → M))
    (hs : ∀ f ∈ s, AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (fun x => (s.map fun f : α → M => f x).prod) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {M : Type u_5} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M] (s : Multiset (α → M)) (hs : ∀ f ∈ s, MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => (Multiset.map (fun f => f x) s).prod) μ
-- Multiset.aestronglyMeasurable_sum
@[to_additive (attr := measurability)]
theorem _root_.Multiset.aestronglyMeasurable_prod (s : Multiset (α → M))
    (hs : ∀ f ∈ s, AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (fun x => (s.map fun f : α → M => f x).prod) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {M : Type u_5} [AddCommMonoid M] [TopologicalSpace M] [ContinuousAdd M] (s : Multiset (α → M)) (hs : ∀ f ∈ s, MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => (Multiset.map (fun f => f x) s).sum) μ
-- Finset.aestronglyMeasurable_sum'
@[to_additive (attr := measurability)]
theorem _root_.Finset.aestronglyMeasurable_prod' {ι : Type*} {f : ι → α → M} (s : Finset ι)
    (hf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ) : AEStronglyMeasurable (∏ i ∈ s, f i) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {M : Type u_5} [AddCommMonoid M] [TopologicalSpace M] [ContinuousAdd M] {ι : Type u_6} {f : ι → α → M} (s : Finset ι) (hf : ∀ i ∈ s, MeasureTheory.AEStronglyMeasurable (f i) μ)
Goal: MeasureTheory.AEStronglyMeasurable (∑ i ∈ s, f i) μ
-- Finset.aestronglyMeasurable_prod'
@[to_additive (attr := measurability)]
theorem _root_.Finset.aestronglyMeasurable_prod' {ι : Type*} {f : ι → α → M} (s : Finset ι)
    (hf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ) : AEStronglyMeasurable (∏ i ∈ s, f i) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {M : Type u_5} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M] {ι : Type u_6} {f : ι → α → M} (s : Finset ι) (hf : ∀ i ∈ s, MeasureTheory.AEStronglyMeasurable (f i) μ)
Goal: MeasureTheory.AEStronglyMeasurable (∏ i ∈ s, f i) μ
-- Finset.aestronglyMeasurable_prod
@[to_additive (attr := measurability)]
theorem _root_.Finset.aestronglyMeasurable_prod {ι : Type*} {f : ι → α → M} (s : Finset ι)
    (hf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ) :
    AEStronglyMeasurable (fun a => ∏ i ∈ s, f i a) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {M : Type u_5} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M] {ι : Type u_6} {f : ι → α → M} (s : Finset ι) (hf : ∀ i ∈ s, MeasureTheory.AEStronglyMeasurable (f i) μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun a => ∏ i ∈ s, f i a) μ
-- Finset.aestronglyMeasurable_sum
@[to_additive (attr := measurability)]
theorem _root_.Finset.aestronglyMeasurable_prod {ι : Type*} {f : ι → α → M} (s : Finset ι)
    (hf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ) :
    AEStronglyMeasurable (fun a => ∏ i ∈ s, f i a) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {M : Type u_5} [AddCommMonoid M] [TopologicalSpace M] [ContinuousAdd M] {ι : Type u_6} {f : ι → α → M} (s : Finset ι) (hf : ∀ i ∈ s, MeasureTheory.AEStronglyMeasurable (f i) μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun a => ∑ i ∈ s, f i a) μ
-- AEMeasurable.aestronglyMeasurable
@[aesop 90% apply (rule_sets := [Measurable])]
theorem _root_.AEMeasurable.aestronglyMeasurable [PseudoMetrizableSpace β] [OpensMeasurableSpace β]
    [SecondCountableTopology β] (hf : AEMeasurable f μ) : AEStronglyMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [MeasurableSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [OpensMeasurableSpace β] [SecondCountableTopology β] (hf : AEMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable f μ
-- aestronglyMeasurable_id
@[measurability]
theorem _root_.aestronglyMeasurable_id {α : Type*} [TopologicalSpace α] [PseudoMetrizableSpace α]
    {_ : MeasurableSpace α} [OpensMeasurableSpace α] [SecondCountableTopology α] {μ : Measure α} :
    AEStronglyMeasurable (id : α → α) μ 
Arguments: {α : Type u_5} [TopologicalSpace α] [TopologicalSpace.PseudoMetrizableSpace α]
Goal: ∀ {x : MeasurableSpace α} [inst : OpensMeasurableSpace α] [inst : SecondCountableTopology α]
  {μ : MeasureTheory.Measure α}, MeasureTheory.AEStronglyMeasurable id μ
-- aestronglyMeasurable_iff_aemeasurable
theorem _root_.aestronglyMeasurable_iff_aemeasurable [PseudoMetrizableSpace β] [BorelSpace β]
    [SecondCountableTopology β] : AEStronglyMeasurable f μ ↔ AEMeasurable f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [MeasurableSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [BorelSpace β] [SecondCountableTopology β]
Goal: MeasureTheory.AEStronglyMeasurable f μ ↔ AEMeasurable f μ
-- MeasureTheory.AEStronglyMeasurable.dist
@[aesop safe 20 apply (rule_sets := [Measurable])]
protected theorem dist {β : Type*} [PseudoMetricSpace β] {f g : α → β}
    (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
    AEStronglyMeasurable (fun x => dist (f x) (g x)) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_5} [PseudoMetricSpace β] {f g : α → β} (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => dist (f x) (g x)) μ
-- MeasureTheory.AEStronglyMeasurable.norm
@[measurability]
protected theorem norm {β : Type*} [SeminormedAddCommGroup β] {f : α → β}
    (hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable (fun x => ‖f x‖) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_5} [SeminormedAddCommGroup β] {f : α → β} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => ‖f x‖) μ
-- MeasureTheory.AEStronglyMeasurable.nnnorm
@[measurability]
protected theorem nnnorm {β : Type*} [SeminormedAddCommGroup β] {f : α → β}
    (hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable (fun x => ‖f x‖₊) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_5} [SeminormedAddCommGroup β] {f : α → β} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => ‖f x‖₊) μ
-- MeasureTheory.AEStronglyMeasurable.ennnorm
@[measurability]
protected theorem ennnorm {β : Type*} [SeminormedAddCommGroup β] {f : α → β}
    (hf : AEStronglyMeasurable f μ) : AEMeasurable (fun a => (‖f a‖₊ : ℝ≥0∞)) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_5} [SeminormedAddCommGroup β] {f : α → β} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: AEMeasurable (fun a => ↑‖f a‖₊) μ