ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasurableSMul.measurable_const_smul
@[to_additive]
class MeasurableSMul (M α : Type*) [SMul M α] [MeasurableSpace M] [MeasurableSpace α] :
    Prop where
  measurable_const_smul : ∀ c : M, Measurable (c • · : α → α)
  measurable_smul_const : ∀ x : α, Measurable (· • x : M → α)
Arguments: {M : Type u_2} {α : Type u_3} [SMul M α] [MeasurableSpace M] [MeasurableSpace α] [self : MeasurableSMul M α] (c : M)
Goal: Measurable fun x => c • x
-- MeasurableVAdd₂.measurable_vadd
class MeasurableVAdd₂ (M α : Type*) [VAdd M α] [MeasurableSpace M] [MeasurableSpace α] :
    Prop where
  measurable_vadd : Measurable (Function.uncurry (· +ᵥ ·) : M × α → α)
Arguments: {M : Type u_2} {α : Type u_3} [VAdd M α] [MeasurableSpace M] [MeasurableSpace α] [self : MeasurableVAdd₂ M α]
Goal: Measurable (Function.uncurry fun x x_1 => x +ᵥ x_1)
-- MeasurableSMul₂.measurable_smul
@[to_additive MeasurableVAdd₂]
class MeasurableSMul₂ (M α : Type*) [SMul M α] [MeasurableSpace M] [MeasurableSpace α] :
    Prop where
  measurable_smul : Measurable (Function.uncurry (· • ·) : M × α → α)
Arguments: {M : Type u_2} {α : Type u_3} [SMul M α] [MeasurableSpace M] [MeasurableSpace α] [self : MeasurableSMul₂ M α]
Goal: Measurable (Function.uncurry fun x x_1 => x • x_1)
-- Measurable.vadd
@[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))]
theorem Measurable.smul [MeasurableSMul₂ M β] (hf : Measurable f) (hg : Measurable g) :
    Measurable fun x => f x • g x 
Arguments: {M : Type u_2} {β : Type u_3} {α : Type u_4} [MeasurableSpace M] [MeasurableSpace β] [VAdd M β] {m : MeasurableSpace α} {f : α → M} {g : α → β} [MeasurableVAdd₂ M β] (hf : Measurable f) (hg : Measurable g)
Goal: Measurable fun x => f x +ᵥ g x
-- Measurable.smul
@[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))]
theorem Measurable.smul [MeasurableSMul₂ M β] (hf : Measurable f) (hg : Measurable g) :
    Measurable fun x => f x • g x 
Arguments: {M : Type u_2} {β : Type u_3} {α : Type u_4} [MeasurableSpace M] [MeasurableSpace β] [SMul M β] {m : MeasurableSpace α} {f : α → M} {g : α → β} [MeasurableSMul₂ M β] (hf : Measurable f) (hg : Measurable g)
Goal: Measurable fun x => f x • g x
-- AEMeasurable.vadd
@[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))]
theorem AEMeasurable.smul [MeasurableSMul₂ M β] {μ : Measure α} (hf : AEMeasurable f μ)
    (hg : AEMeasurable g μ) : AEMeasurable (fun x => f x • g x) μ 
Arguments: {M : Type u_2} {β : Type u_3} {α : Type u_4} [MeasurableSpace M] [MeasurableSpace β] [VAdd M β] {m : MeasurableSpace α} {f : α → M} {g : α → β} [MeasurableVAdd₂ M β] {μ : MeasureTheory.Measure α} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: AEMeasurable (fun x => f x +ᵥ g x) μ
-- AEMeasurable.smul
@[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))]
theorem AEMeasurable.smul [MeasurableSMul₂ M β] {μ : Measure α} (hf : AEMeasurable f μ)
    (hg : AEMeasurable g μ) : AEMeasurable (fun x => f x • g x) μ 
Arguments: {M : Type u_2} {β : Type u_3} {α : Type u_4} [MeasurableSpace M] [MeasurableSpace β] [SMul M β] {m : MeasurableSpace α} {f : α → M} {g : α → β} [MeasurableSMul₂ M β] {μ : MeasureTheory.Measure α} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: AEMeasurable (fun x => f x • g x) μ
-- Measurable.vadd_const
@[to_additive (attr := measurability)]
theorem Measurable.smul_const (hf : Measurable f) (y : β) : Measurable fun x => f x • y 
Arguments: {M : Type u_2} {β : Type u_3} {α : Type u_4} [MeasurableSpace M] [MeasurableSpace β] [VAdd M β] {m : MeasurableSpace α} {f : α → M} [MeasurableVAdd M β] (hf : Measurable f) (y : β)
Goal: Measurable fun x => f x +ᵥ y
-- Measurable.smul_const
@[to_additive (attr := measurability)]
theorem Measurable.smul_const (hf : Measurable f) (y : β) : Measurable fun x => f x • y 
Arguments: {M : Type u_2} {β : Type u_3} {α : Type u_4} [MeasurableSpace M] [MeasurableSpace β] [SMul M β] {m : MeasurableSpace α} {f : α → M} [MeasurableSMul M β] (hf : Measurable f) (y : β)
Goal: Measurable fun x => f x • y
-- AEMeasurable.vadd_const
@[to_additive (attr := measurability)]
theorem AEMeasurable.smul_const (hf : AEMeasurable f μ) (y : β) :
    AEMeasurable (fun x => f x • y) μ 
Arguments: {M : Type u_2} {β : Type u_3} {α : Type u_4} [MeasurableSpace M] [MeasurableSpace β] [VAdd M β] {m : MeasurableSpace α} {f : α → M} [MeasurableVAdd M β] {μ : MeasureTheory.Measure α} (hf : AEMeasurable f μ) (y : β)
Goal: AEMeasurable (fun x => f x +ᵥ y) μ
-- AEMeasurable.smul_const
@[to_additive (attr := measurability)]
theorem AEMeasurable.smul_const (hf : AEMeasurable f μ) (y : β) :
    AEMeasurable (fun x => f x • y) μ 
Arguments: {M : Type u_2} {β : Type u_3} {α : Type u_4} [MeasurableSpace M] [MeasurableSpace β] [SMul M β] {m : MeasurableSpace α} {f : α → M} [MeasurableSMul M β] {μ : MeasureTheory.Measure α} (hf : AEMeasurable f μ) (y : β)
Goal: AEMeasurable (fun x => f x • y) μ
-- Measurable.const_vadd'
@[to_additive (attr := measurability)]
theorem Measurable.const_smul' (hg : Measurable g) (c : M) : Measurable fun x => c • g x 
Arguments: {M : Type u_2} {β : Type u_3} {α : Type u_4} [MeasurableSpace M] [MeasurableSpace β] [VAdd M β] {m : MeasurableSpace α} {g : α → β} [MeasurableVAdd M β] (hg : Measurable g) (c : M)
Goal: Measurable fun x => c +ᵥ g x
-- Measurable.const_smul'
@[to_additive (attr := measurability)]
theorem Measurable.const_smul' (hg : Measurable g) (c : M) : Measurable fun x => c • g x 
Arguments: {M : Type u_2} {β : Type u_3} {α : Type u_4} [MeasurableSpace M] [MeasurableSpace β] [SMul M β] {m : MeasurableSpace α} {g : α → β} [MeasurableSMul M β] (hg : Measurable g) (c : M)
Goal: Measurable fun x => c • g x
-- Measurable.const_smul
@[to_additive (attr := measurability)]
theorem Measurable.const_smul (hg : Measurable g) (c : M) : Measurable (c • g) 
Arguments: {M : Type u_2} {β : Type u_3} {α : Type u_4} [MeasurableSpace M] [MeasurableSpace β] [SMul M β] {m : MeasurableSpace α} {g : α → β} [MeasurableSMul M β] (hg : Measurable g) (c : M)
Goal: Measurable (c • g)
-- Measurable.const_vadd
@[to_additive (attr := measurability)]
theorem Measurable.const_smul (hg : Measurable g) (c : M) : Measurable (c • g) 
Arguments: {M : Type u_2} {β : Type u_3} {α : Type u_4} [MeasurableSpace M] [MeasurableSpace β] [VAdd M β] {m : MeasurableSpace α} {g : α → β} [MeasurableVAdd M β] (hg : Measurable g) (c : M)
Goal: Measurable (c +ᵥ g)
-- AEMeasurable.const_smul'
@[to_additive (attr := measurability)]
theorem AEMeasurable.const_smul' (hg : AEMeasurable g μ) (c : M) :
    AEMeasurable (fun x => c • g x) μ 
Arguments: {M : Type u_2} {β : Type u_3} {α : Type u_4} [MeasurableSpace M] [MeasurableSpace β] [SMul M β] {m : MeasurableSpace α} {g : α → β} [MeasurableSMul M β] {μ : MeasureTheory.Measure α} (hg : AEMeasurable g μ) (c : M)
Goal: AEMeasurable (fun x => c • g x) μ
-- AEMeasurable.const_vadd'
@[to_additive (attr := measurability)]
theorem AEMeasurable.const_smul' (hg : AEMeasurable g μ) (c : M) :
    AEMeasurable (fun x => c • g x) μ 
Arguments: {M : Type u_2} {β : Type u_3} {α : Type u_4} [MeasurableSpace M] [MeasurableSpace β] [VAdd M β] {m : MeasurableSpace α} {g : α → β} [MeasurableVAdd M β] {μ : MeasureTheory.Measure α} (hg : AEMeasurable g μ) (c : M)
Goal: AEMeasurable (fun x => c +ᵥ g x) μ
-- AEMeasurable.const_smul
@[to_additive (attr := measurability)]
theorem AEMeasurable.const_smul (hf : AEMeasurable g μ) (c : M) : AEMeasurable (c • g) μ 
Arguments: {M : Type u_2} {β : Type u_3} {α : Type u_4} [MeasurableSpace M] [MeasurableSpace β] [SMul M β] {m : MeasurableSpace α} {g : α → β} [MeasurableSMul M β] {μ : MeasureTheory.Measure α} (hf : AEMeasurable g μ) (c : M)
Goal: AEMeasurable (c • g) μ
-- AEMeasurable.const_vadd
@[to_additive (attr := measurability)]
theorem AEMeasurable.const_smul (hf : AEMeasurable g μ) (c : M) : AEMeasurable (c • g) μ 
Arguments: {M : Type u_2} {β : Type u_3} {α : Type u_4} [MeasurableSpace M] [MeasurableSpace β] [VAdd M β] {m : MeasurableSpace α} {g : α → β} [MeasurableVAdd M β] {μ : MeasureTheory.Measure α} (hf : AEMeasurable g μ) (c : M)
Goal: AEMeasurable (c +ᵥ g) μ
-- measurable_const_smul_iff
@[to_additive]
theorem measurable_const_smul_iff (c : G) : (Measurable fun x => c • f x) ↔ Measurable f 
Arguments: {β : Type u_3} {α : Type u_4} [MeasurableSpace β] [MeasurableSpace α] {f : α → β} {G : Type u_5} [Group G] [MeasurableSpace G] [MulAction G β] [MeasurableSMul G β] (c : G)
Goal: (Measurable fun x => c • f x) ↔ Measurable f
-- measurable_const_vadd_iff
@[to_additive]
theorem measurable_const_smul_iff (c : G) : (Measurable fun x => c • f x) ↔ Measurable f 
Arguments: {β : Type u_3} {α : Type u_4} [MeasurableSpace β] [MeasurableSpace α] {f : α → β} {G : Type u_5} [AddGroup G] [MeasurableSpace G] [AddAction G β] [MeasurableVAdd G β] (c : G)
Goal: (Measurable fun x => c +ᵥ f x) ↔ Measurable f
-- aemeasurable_const_smul_iff
@[to_additive]
theorem aemeasurable_const_smul_iff (c : G) :
    AEMeasurable (fun x => c • f x) μ ↔ AEMeasurable f μ 
Arguments: {β : Type u_3} {α : Type u_4} [MeasurableSpace β] [MeasurableSpace α] {f : α → β} {μ : MeasureTheory.Measure α} {G : Type u_5} [Group G] [MeasurableSpace G] [MulAction G β] [MeasurableSMul G β] (c : G)
Goal: AEMeasurable (fun x => c • f x) μ ↔ AEMeasurable f μ
-- aemeasurable_const_vadd_iff
@[to_additive]
theorem aemeasurable_const_smul_iff (c : G) :
    AEMeasurable (fun x => c • f x) μ ↔ AEMeasurable f μ 
Arguments: {β : Type u_3} {α : Type u_4} [MeasurableSpace β] [MeasurableSpace α] {f : α → β} {μ : MeasureTheory.Measure α} {G : Type u_5} [AddGroup G] [MeasurableSpace G] [AddAction G β] [MeasurableVAdd G β] (c : G)
Goal: AEMeasurable (fun x => c +ᵥ f x) μ ↔ AEMeasurable f μ
-- IsAddUnit.measurable_const_vadd_iff
@[to_additive]
nonrec theorem IsUnit.measurable_const_smul_iff {c : M} (hc : IsUnit c) :
    (Measurable fun x => c • f x) ↔ Measurable f 
Arguments: {M : Type u_2} {β : Type u_3} {α : Type u_4} [MeasurableSpace M] [MeasurableSpace β] [AddMonoid M] [AddAction M β] [MeasurableVAdd M β] [MeasurableSpace α] {f : α → β} {c : M} (hc : IsAddUnit c)
Goal: (Measurable fun x => c +ᵥ f x) ↔ Measurable f
-- IsUnit.measurable_const_smul_iff
@[to_additive]
nonrec theorem IsUnit.measurable_const_smul_iff {c : M} (hc : IsUnit c) :
    (Measurable fun x => c • f x) ↔ Measurable f 
Arguments: {M : Type u_2} {β : Type u_3} {α : Type u_4} [MeasurableSpace M] [MeasurableSpace β] [Monoid M] [MulAction M β] [MeasurableSMul M β] [MeasurableSpace α] {f : α → β} {c : M} (hc : IsUnit c)
Goal: (Measurable fun x => c • f x) ↔ Measurable f
-- IsAddUnit.aemeasurable_const_vadd_iff
@[to_additive]
nonrec theorem IsUnit.aemeasurable_const_smul_iff {c : M} (hc : IsUnit c) :
    AEMeasurable (fun x => c • f x) μ ↔ AEMeasurable f μ 
Arguments: {M : Type u_2} {β : Type u_3} {α : Type u_4} [MeasurableSpace M] [MeasurableSpace β] [AddMonoid M] [AddAction M β] [MeasurableVAdd M β] [MeasurableSpace α] {f : α → β} {μ : MeasureTheory.Measure α} {c : M} (hc : IsAddUnit c)
Goal: AEMeasurable (fun x => c +ᵥ f x) μ ↔ AEMeasurable f μ
-- IsUnit.aemeasurable_const_smul_iff
@[to_additive]
nonrec theorem IsUnit.aemeasurable_const_smul_iff {c : M} (hc : IsUnit c) :
    AEMeasurable (fun x => c • f x) μ ↔ AEMeasurable f μ 
Arguments: {M : Type u_2} {β : Type u_3} {α : Type u_4} [MeasurableSpace M] [MeasurableSpace β] [Monoid M] [MulAction M β] [MeasurableSMul M β] [MeasurableSpace α] {f : α → β} {μ : MeasureTheory.Measure α} {c : M} (hc : IsUnit c)
Goal: AEMeasurable (fun x => c • f x) μ ↔ AEMeasurable f μ
-- measurable_const_smul_iff₀
theorem measurable_const_smul_iff₀ {c : G₀} (hc : c ≠ 0) :
    (Measurable fun x => c • f x) ↔ Measurable f 
Arguments: {β : Type u_3} {α : Type u_4} [MeasurableSpace β] [MeasurableSpace α] {f : α → β} {G₀ : Type u_6} [GroupWithZero G₀] [MeasurableSpace G₀] [MulAction G₀ β] [MeasurableSMul G₀ β] {c : G₀} (hc : c ≠ 0)
Goal: (Measurable fun x => c • f x) ↔ Measurable f
-- aemeasurable_const_smul_iff₀
theorem aemeasurable_const_smul_iff₀ {c : G₀} (hc : c ≠ 0) :
    AEMeasurable (fun x => c • f x) μ ↔ AEMeasurable f μ 
Arguments: {β : Type u_3} {α : Type u_4} [MeasurableSpace β] [MeasurableSpace α] {f : α → β} {μ : MeasureTheory.Measure α} {G₀ : Type u_6} [GroupWithZero G₀] [MeasurableSpace G₀] [MulAction G₀ β] [MeasurableSMul G₀ β] {c : G₀} (hc : c ≠ 0)
Goal: AEMeasurable (fun x => c • f x) μ ↔ AEMeasurable f μ
-- measurable_add_op
@[to_additive]
theorem measurable_mul_op {α : Type*} [MeasurableSpace α] : Measurable (op : α → αᵐᵒᵖ) 
Arguments: {α : Type u_2} [MeasurableSpace α]
Goal: Measurable AddOpposite.op
-- measurable_mul_op
@[to_additive]
theorem measurable_mul_op {α : Type*} [MeasurableSpace α] : Measurable (op : α → αᵐᵒᵖ) 
Arguments: {α : Type u_2} [MeasurableSpace α]
Goal: Measurable MulOpposite.op
-- measurable_add_unop
@[to_additive]
theorem measurable_mul_unop {α : Type*} [MeasurableSpace α] : Measurable (unop : αᵐᵒᵖ → α) 
Arguments: {α : Type u_2} [MeasurableSpace α]
Goal: Measurable AddOpposite.unop
-- measurable_mul_unop
@[to_additive]
theorem measurable_mul_unop {α : Type*} [MeasurableSpace α] : Measurable (unop : αᵐᵒᵖ → α) 
Arguments: {α : Type u_2} [MeasurableSpace α]
Goal: Measurable MulOpposite.unop
-- List.measurable_sum'
@[to_additive (attr := measurability)]
theorem List.measurable_prod' (l : List (α → M)) (hl : ∀ f ∈ l, Measurable f) :
    Measurable l.prod 
Arguments: {M : Type u_2} {α : Type u_3} [AddMonoid M] [MeasurableSpace M] [MeasurableAdd₂ M] {m : MeasurableSpace α} (l : List (α → M)) (hl : ∀ f ∈ l, Measurable f)
Goal: Measurable l.sum
-- List.measurable_prod'
@[to_additive (attr := measurability)]
theorem List.measurable_prod' (l : List (α → M)) (hl : ∀ f ∈ l, Measurable f) :
    Measurable l.prod 
Arguments: {M : Type u_2} {α : Type u_3} [Monoid M] [MeasurableSpace M] [MeasurableMul₂ M] {m : MeasurableSpace α} (l : List (α → M)) (hl : ∀ f ∈ l, Measurable f)
Goal: Measurable l.prod
-- List.aemeasurable_sum'
@[to_additive (attr := measurability)]
theorem List.aemeasurable_prod' (l : List (α → M)) (hl : ∀ f ∈ l, AEMeasurable f μ) :
    AEMeasurable l.prod μ 
Arguments: {M : Type u_2} {α : Type u_3} [AddMonoid M] [MeasurableSpace M] [MeasurableAdd₂ M] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (l : List (α → M)) (hl : ∀ f ∈ l, AEMeasurable f μ)
Goal: AEMeasurable l.sum μ
-- List.aemeasurable_prod'
@[to_additive (attr := measurability)]
theorem List.aemeasurable_prod' (l : List (α → M)) (hl : ∀ f ∈ l, AEMeasurable f μ) :
    AEMeasurable l.prod μ 
Arguments: {M : Type u_2} {α : Type u_3} [Monoid M] [MeasurableSpace M] [MeasurableMul₂ M] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (l : List (α → M)) (hl : ∀ f ∈ l, AEMeasurable f μ)
Goal: AEMeasurable l.prod μ
-- List.measurable_sum
@[to_additive (attr := measurability)]
theorem List.measurable_prod (l : List (α → M)) (hl : ∀ f ∈ l, Measurable f) :
    Measurable fun x => (l.map fun f : α → M => f x).prod 
Arguments: {M : Type u_2} {α : Type u_3} [AddMonoid M] [MeasurableSpace M] [MeasurableAdd₂ M] {m : MeasurableSpace α} (l : List (α → M)) (hl : ∀ f ∈ l, Measurable f)
Goal: Measurable fun x => (List.map (fun f => f x) l).sum
-- List.measurable_prod
@[to_additive (attr := measurability)]
theorem List.measurable_prod (l : List (α → M)) (hl : ∀ f ∈ l, Measurable f) :
    Measurable fun x => (l.map fun f : α → M => f x).prod 
Arguments: {M : Type u_2} {α : Type u_3} [Monoid M] [MeasurableSpace M] [MeasurableMul₂ M] {m : MeasurableSpace α} (l : List (α → M)) (hl : ∀ f ∈ l, Measurable f)
Goal: Measurable fun x => (List.map (fun f => f x) l).prod
-- List.aemeasurable_prod
@[to_additive (attr := measurability)]
theorem List.aemeasurable_prod (l : List (α → M)) (hl : ∀ f ∈ l, AEMeasurable f μ) :
    AEMeasurable (fun x => (l.map fun f : α → M => f x).prod) μ 
Arguments: {M : Type u_2} {α : Type u_3} [Monoid M] [MeasurableSpace M] [MeasurableMul₂ M] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (l : List (α → M)) (hl : ∀ f ∈ l, AEMeasurable f μ)
Goal: AEMeasurable (fun x => (List.map (fun f => f x) l).prod) μ
-- List.aemeasurable_sum
@[to_additive (attr := measurability)]
theorem List.aemeasurable_prod (l : List (α → M)) (hl : ∀ f ∈ l, AEMeasurable f μ) :
    AEMeasurable (fun x => (l.map fun f : α → M => f x).prod) μ 
Arguments: {M : Type u_2} {α : Type u_3} [AddMonoid M] [MeasurableSpace M] [MeasurableAdd₂ M] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (l : List (α → M)) (hl : ∀ f ∈ l, AEMeasurable f μ)
Goal: AEMeasurable (fun x => (List.map (fun f => f x) l).sum) μ
-- Multiset.measurable_prod'
@[to_additive (attr := measurability)]
theorem Multiset.measurable_prod' (l : Multiset (α → M)) (hl : ∀ f ∈ l, Measurable f) :
    Measurable l.prod 
Arguments: {M : Type u_2} {α : Type u_4} [CommMonoid M] [MeasurableSpace M] [MeasurableMul₂ M] {m : MeasurableSpace α} (l : Multiset (α → M)) (hl : ∀ f ∈ l, Measurable f)
Goal: Measurable l.prod
-- Multiset.measurable_sum'
@[to_additive (attr := measurability)]
theorem Multiset.measurable_prod' (l : Multiset (α → M)) (hl : ∀ f ∈ l, Measurable f) :
    Measurable l.prod 
Arguments: {M : Type u_2} {α : Type u_4} [AddCommMonoid M] [MeasurableSpace M] [MeasurableAdd₂ M] {m : MeasurableSpace α} (l : Multiset (α → M)) (hl : ∀ f ∈ l, Measurable f)
Goal: Measurable l.sum
-- Multiset.aemeasurable_prod'
@[to_additive (attr := measurability)]
theorem Multiset.aemeasurable_prod' (l : Multiset (α → M)) (hl : ∀ f ∈ l, AEMeasurable f μ) :
    AEMeasurable l.prod μ 
Arguments: {M : Type u_2} {α : Type u_4} [CommMonoid M] [MeasurableSpace M] [MeasurableMul₂ M] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (l : Multiset (α → M)) (hl : ∀ f ∈ l, AEMeasurable f μ)
Goal: AEMeasurable l.prod μ
-- Multiset.aemeasurable_sum'
@[to_additive (attr := measurability)]
theorem Multiset.aemeasurable_prod' (l : Multiset (α → M)) (hl : ∀ f ∈ l, AEMeasurable f μ) :
    AEMeasurable l.prod μ 
Arguments: {M : Type u_2} {α : Type u_4} [AddCommMonoid M] [MeasurableSpace M] [MeasurableAdd₂ M] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (l : Multiset (α → M)) (hl : ∀ f ∈ l, AEMeasurable f μ)
Goal: AEMeasurable l.sum μ
-- Multiset.measurable_sum
@[to_additive (attr := measurability)]
theorem Multiset.measurable_prod (s : Multiset (α → M)) (hs : ∀ f ∈ s, Measurable f) :
    Measurable fun x => (s.map fun f : α → M => f x).prod 
Arguments: {M : Type u_2} {α : Type u_4} [AddCommMonoid M] [MeasurableSpace M] [MeasurableAdd₂ M] {m : MeasurableSpace α} (s : Multiset (α → M)) (hs : ∀ f ∈ s, Measurable f)
Goal: Measurable fun x => (Multiset.map (fun f => f x) s).sum
-- Multiset.measurable_prod
@[to_additive (attr := measurability)]
theorem Multiset.measurable_prod (s : Multiset (α → M)) (hs : ∀ f ∈ s, Measurable f) :
    Measurable fun x => (s.map fun f : α → M => f x).prod 
Arguments: {M : Type u_2} {α : Type u_4} [CommMonoid M] [MeasurableSpace M] [MeasurableMul₂ M] {m : MeasurableSpace α} (s : Multiset (α → M)) (hs : ∀ f ∈ s, Measurable f)
Goal: Measurable fun x => (Multiset.map (fun f => f x) s).prod
-- Multiset.aemeasurable_prod
@[to_additive (attr := measurability)]
theorem Multiset.aemeasurable_prod (s : Multiset (α → M)) (hs : ∀ f ∈ s, AEMeasurable f μ) :
    AEMeasurable (fun x => (s.map fun f : α → M => f x).prod) μ 
Arguments: {M : Type u_2} {α : Type u_4} [CommMonoid M] [MeasurableSpace M] [MeasurableMul₂ M] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Multiset (α → M)) (hs : ∀ f ∈ s, AEMeasurable f μ)
Goal: AEMeasurable (fun x => (Multiset.map (fun f => f x) s).prod) μ
-- Multiset.aemeasurable_sum
@[to_additive (attr := measurability)]
theorem Multiset.aemeasurable_prod (s : Multiset (α → M)) (hs : ∀ f ∈ s, AEMeasurable f μ) :
    AEMeasurable (fun x => (s.map fun f : α → M => f x).prod) μ 
Arguments: {M : Type u_2} {α : Type u_4} [AddCommMonoid M] [MeasurableSpace M] [MeasurableAdd₂ M] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Multiset (α → M)) (hs : ∀ f ∈ s, AEMeasurable f μ)
Goal: AEMeasurable (fun x => (Multiset.map (fun f => f x) s).sum) μ
-- Finset.measurable_sum'
@[to_additive (attr := measurability)]
theorem Finset.measurable_prod' (s : Finset ι) (hf : ∀ i ∈ s, Measurable (f i)) :
    Measurable (∏ i ∈ s, f i) 
Arguments: {M : Type u_2} {ι : Type u_3} {α : Type u_4} [AddCommMonoid M] [MeasurableSpace M] [MeasurableAdd₂ M] {m : MeasurableSpace α} {f : ι → α → M} (s : Finset ι) (hf : ∀ i ∈ s, Measurable (f i))
Goal: Measurable (∑ i ∈ s, f i)
-- Finset.measurable_prod'
@[to_additive (attr := measurability)]
theorem Finset.measurable_prod' (s : Finset ι) (hf : ∀ i ∈ s, Measurable (f i)) :
    Measurable (∏ i ∈ s, f i) 
Arguments: {M : Type u_2} {ι : Type u_3} {α : Type u_4} [CommMonoid M] [MeasurableSpace M] [MeasurableMul₂ M] {m : MeasurableSpace α} {f : ι → α → M} (s : Finset ι) (hf : ∀ i ∈ s, Measurable (f i))
Goal: Measurable (∏ i ∈ s, f i)
-- Finset.measurable_prod
@[to_additive (attr := measurability)]
theorem Finset.measurable_prod (s : Finset ι) (hf : ∀ i ∈ s, Measurable (f i)) :
    Measurable fun a => ∏ i ∈ s, f i a 
Arguments: {M : Type u_2} {ι : Type u_3} {α : Type u_4} [CommMonoid M] [MeasurableSpace M] [MeasurableMul₂ M] {m : MeasurableSpace α} {f : ι → α → M} (s : Finset ι) (hf : ∀ i ∈ s, Measurable (f i))
Goal: Measurable fun a => ∏ i ∈ s, f i a
-- Finset.measurable_sum
@[to_additive (attr := measurability)]
theorem Finset.measurable_prod (s : Finset ι) (hf : ∀ i ∈ s, Measurable (f i)) :
    Measurable fun a => ∏ i ∈ s, f i a 
Arguments: {M : Type u_2} {ι : Type u_3} {α : Type u_4} [AddCommMonoid M] [MeasurableSpace M] [MeasurableAdd₂ M] {m : MeasurableSpace α} {f : ι → α → M} (s : Finset ι) (hf : ∀ i ∈ s, Measurable (f i))
Goal: Measurable fun a => ∑ i ∈ s, f i a
-- Finset.aemeasurable_prod'
@[to_additive (attr := measurability)]
theorem Finset.aemeasurable_prod' (s : Finset ι) (hf : ∀ i ∈ s, AEMeasurable (f i) μ) :
    AEMeasurable (∏ i ∈ s, f i) μ 
Arguments: {M : Type u_2} {ι : Type u_3} {α : Type u_4} [CommMonoid M] [MeasurableSpace M] [MeasurableMul₂ M] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ι → α → M} (s : Finset ι) (hf : ∀ i ∈ s, AEMeasurable (f i) μ)
Goal: AEMeasurable (∏ i ∈ s, f i) μ
-- Finset.aemeasurable_sum'
@[to_additive (attr := measurability)]
theorem Finset.aemeasurable_prod' (s : Finset ι) (hf : ∀ i ∈ s, AEMeasurable (f i) μ) :
    AEMeasurable (∏ i ∈ s, f i) μ 
Arguments: {M : Type u_2} {ι : Type u_3} {α : Type u_4} [AddCommMonoid M] [MeasurableSpace M] [MeasurableAdd₂ M] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ι → α → M} (s : Finset ι) (hf : ∀ i ∈ s, AEMeasurable (f i) μ)
Goal: AEMeasurable (∑ i ∈ s, f i) μ
-- Finset.aemeasurable_prod
@[to_additive (attr := measurability)]
theorem Finset.aemeasurable_prod (s : Finset ι) (hf : ∀ i ∈ s, AEMeasurable (f i) μ) :
    AEMeasurable (fun a => ∏ i ∈ s, f i a) μ 
Arguments: {M : Type u_2} {ι : Type u_3} {α : Type u_4} [CommMonoid M] [MeasurableSpace M] [MeasurableMul₂ M] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ι → α → M} (s : Finset ι) (hf : ∀ i ∈ s, AEMeasurable (f i) μ)
Goal: AEMeasurable (fun a => ∏ i ∈ s, f i a) μ
-- Finset.aemeasurable_sum
@[to_additive (attr := measurability)]
theorem Finset.aemeasurable_prod (s : Finset ι) (hf : ∀ i ∈ s, AEMeasurable (f i) μ) :
    AEMeasurable (fun a => ∏ i ∈ s, f i a) μ 
Arguments: {M : Type u_2} {ι : Type u_3} {α : Type u_4} [AddCommMonoid M] [MeasurableSpace M] [MeasurableAdd₂ M] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ι → α → M} (s : Finset ι) (hf : ∀ i ∈ s, AEMeasurable (f i) μ)
Goal: AEMeasurable (fun a => ∑ i ∈ s, f i a) μ
-- MeasureTheory.withDensity_apply
@[simp]
theorem withDensity_apply (f : α → ℝ≥0∞) {s : Set α} (hs : MeasurableSet s) :
    μ.withDensity f s = ∫⁻ a in s, f a ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ℝ≥0∞) {s : Set α} (hs : MeasurableSet s)
Goal: (μ.withDensity f) s = ∫⁻ (a : α) in s, f a ∂μ
-- MeasureTheory.withDensity_apply_le
theorem withDensity_apply_le (f : α → ℝ≥0∞) (s : Set α) :
    ∫⁻ a in s, f a ∂μ ≤ μ.withDensity f s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ℝ≥0∞) (s : Set α)
Goal: ∫⁻ (a : α) in s, f a ∂μ ≤ (μ.withDensity f) s