ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- generateFrom_pi_eq
theorem generateFrom_pi_eq {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsCountablySpanning (C i)) :
    (@MeasurableSpace.pi _ _ fun i => generateFrom (C i)) =
    generateFrom (pi univ '' pi univ C) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Finite ι] {C : (i : ι) → Set (Set (α i))} (hC : ∀ (i : ι), IsCountablySpanning (C i))
Goal: MeasurableSpace.pi = MeasurableSpace.generateFrom (Set.univ.pi '' Set.univ.pi C)
-- generateFrom_eq_pi
theorem generateFrom_eq_pi [h : ∀ i, MeasurableSpace (α i)] {C : ∀ i, Set (Set (α i))}
    (hC : ∀ i, generateFrom (C i) = h i) (h2C : ∀ i, IsCountablySpanning (C i)) :
    generateFrom (pi univ '' pi univ C) = MeasurableSpace.pi 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Finite ι] [h : (i : ι) → MeasurableSpace (α i)] {C : (i : ι) → Set (Set (α i))} (hC : ∀ (i : ι), MeasurableSpace.generateFrom (C i) = h i) (h2C : ∀ (i : ι), IsCountablySpanning (C i))
Goal: MeasurableSpace.generateFrom (Set.univ.pi '' Set.univ.pi C) = MeasurableSpace.pi
-- generateFrom_pi
theorem generateFrom_pi [∀ i, MeasurableSpace (α i)] :
    generateFrom (pi univ '' pi univ fun i => { s : Set (α i) | MeasurableSet s }) =
      MeasurableSpace.pi 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Finite ι] [(i : ι) → MeasurableSpace (α i)]
Goal: MeasurableSpace.generateFrom (Set.univ.pi '' Set.univ.pi fun i => {s | MeasurableSet s}) = MeasurableSpace.pi
-- MeasureTheory.piPremeasure_pi
theorem piPremeasure_pi {s : ∀ i, Set (α i)} (hs : (pi univ s).Nonempty) :
    piPremeasure m (pi univ s) = ∏ i, m i (s i) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] {m : (i : ι) → MeasureTheory.OuterMeasure (α i)} {s : (i : ι) → Set (α i)} (hs : (Set.univ.pi s).Nonempty)
Goal: MeasureTheory.piPremeasure m (Set.univ.pi s) = ∏ i : ι, (m i) (s i)
-- MeasureTheory.piPremeasure_pi'
theorem piPremeasure_pi' {s : ∀ i, Set (α i)} : piPremeasure m (pi univ s) = ∏ i, m i (s i) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] {m : (i : ι) → MeasureTheory.OuterMeasure (α i)} {s : (i : ι) → Set (α i)}
Goal: MeasureTheory.piPremeasure m (Set.univ.pi s) = ∏ i : ι, (m i) (s i)
-- MeasureTheory.piPremeasure_pi_mono
theorem piPremeasure_pi_mono {s t : Set (∀ i, α i)} (h : s ⊆ t) :
    piPremeasure m s ≤ piPremeasure m t 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] {m : (i : ι) → MeasureTheory.OuterMeasure (α i)} {s t : Set ((i : ι) → α i)} (h : s ⊆ t)
Goal: MeasureTheory.piPremeasure m s ≤ MeasureTheory.piPremeasure m t
-- MeasureTheory.piPremeasure_pi_eval
theorem piPremeasure_pi_eval {s : Set (∀ i, α i)} :
    piPremeasure m (pi univ fun i => eval i '' s) = piPremeasure m s 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] {m : (i : ι) → MeasureTheory.OuterMeasure (α i)} {s : Set ((i : ι) → α i)}
Goal: MeasureTheory.piPremeasure m (Set.univ.pi fun i => Function.eval i '' s) = MeasureTheory.piPremeasure m s
-- MeasureTheory.OuterMeasure.pi_pi_le
theorem pi_pi_le (m : ∀ i, OuterMeasure (α i)) (s : ∀ i, Set (α i)) :
    OuterMeasure.pi m (pi univ s) ≤ ∏ i, m i (s i) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] (m : (i : ι) → MeasureTheory.OuterMeasure (α i)) (s : (i : ι) → Set (α i))
Goal: (MeasureTheory.OuterMeasure.pi m) (Set.univ.pi s) ≤ ∏ i : ι, (m i) (s i)
-- MeasureTheory.OuterMeasure.le_pi
theorem le_pi {m : ∀ i, OuterMeasure (α i)} {n : OuterMeasure (∀ i, α i)} :
    n ≤ OuterMeasure.pi m ↔
      ∀ s : ∀ i, Set (α i), (pi univ s).Nonempty → n (pi univ s) ≤ ∏ i, m i (s i) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] {m : (i : ι) → MeasureTheory.OuterMeasure (α i)} {n : MeasureTheory.OuterMeasure ((i : ι) → α i)}
Goal: n ≤ MeasureTheory.OuterMeasure.pi m ↔
  ∀ (s : (i : ι) → Set (α i)), (Set.univ.pi s).Nonempty → n (Set.univ.pi s) ≤ ∏ i : ι, (m i) (s i)
-- MeasureTheory.Measure.tprod_nil
@[simp]
theorem tprod_nil (μ : ∀ i, Measure (π i)) : Measure.tprod [] μ = dirac PUnit.unit 
Arguments: {δ : Type u_4} {π : δ → Type u_5} [(x : δ) → MeasurableSpace (π x)] (μ : (i : δ) → MeasureTheory.Measure (π i))
Goal: MeasureTheory.Measure.tprod [] μ = MeasureTheory.Measure.dirac PUnit.unit
-- MeasureTheory.Measure.tprod_cons
@[simp]
theorem tprod_cons (i : δ) (l : List δ) (μ : ∀ i, Measure (π i)) :
    Measure.tprod (i :: l) μ = (μ i).prod (Measure.tprod l μ) 
Arguments: {δ : Type u_4} {π : δ → Type u_5} [(x : δ) → MeasurableSpace (π x)] (i : δ) (l : List δ) (μ : (i : δ) → MeasureTheory.Measure (π i))
Goal: MeasureTheory.Measure.tprod (i :: l) μ = (μ i).prod (MeasureTheory.Measure.tprod l μ)
-- MeasureTheory.Measure.tprod_tprod
theorem tprod_tprod (l : List δ) (μ : ∀ i, Measure (π i)) [∀ i, SigmaFinite (μ i)]
    (s : ∀ i, Set (π i)) :
    Measure.tprod l μ (Set.tprod l s) = (l.map fun i => (μ i) (s i)).prod 
Arguments: {δ : Type u_4} {π : δ → Type u_5} [(x : δ) → MeasurableSpace (π x)] (l : List δ) (μ : (i : δ) → MeasureTheory.Measure (π i)) [∀ (i : δ), MeasureTheory.SigmaFinite (μ i)] (s : (i : δ) → Set (π i))
Goal: (MeasureTheory.Measure.tprod l μ) (Set.tprod l s) = (List.map (fun i => (μ i) (s i)) l).prod
-- MeasureTheory.Measure.pi'_pi
theorem pi'_pi [∀ i, SigmaFinite (μ i)] (s : ∀ i, Set (α i)) :
    pi' μ (pi univ s) = ∏ i, μ i (s i) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [Encodable ι] [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] (s : (i : ι) → Set (α i))
Goal: (MeasureTheory.Measure.pi' μ) (Set.univ.pi s) = ∏ i : ι, (μ i) (s i)
-- MeasureTheory.Measure.pi_caratheodory
theorem pi_caratheodory :
    MeasurableSpace.pi ≤ (OuterMeasure.pi fun i => (μ i).toOuterMeasure).caratheodory 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i))
Goal: MeasurableSpace.pi ≤ (MeasureTheory.OuterMeasure.pi fun i => (μ i).toOuterMeasure).caratheodory
-- MeasureTheory.Measure.pi_def
protected irreducible_def pi : Measure (∀ i, α i) 
Arguments: {ι : Type u_4} {α : ι → Type u_5} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i))
Goal: MeasureTheory.Measure.pi μ = (MeasureTheory.OuterMeasure.pi fun i => (μ i).toOuterMeasure).toMeasure ⋯
-- MeasureTheory.Measure.pi_pi_aux
theorem pi_pi_aux [∀ i, SigmaFinite (μ i)] (s : ∀ i, Set (α i)) (hs : ∀ i, MeasurableSet (s i)) :
    Measure.pi μ (pi univ s) = ∏ i, μ i (s i) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] (s : (i : ι) → Set (α i)) (hs : ∀ (i : ι), MeasurableSet (s i))
Goal: (MeasureTheory.Measure.pi μ) (Set.univ.pi s) = ∏ i : ι, (μ i) (s i)
-- MeasureTheory.Measure.pi_eq_generateFrom
theorem pi_eq_generateFrom {C : ∀ i, Set (Set (α i))}
    (hC : ∀ i, generateFrom (C i) = by apply_assumption) (h2C : ∀ i, IsPiSystem (C i))
    (h3C : ∀ i, (μ i).FiniteSpanningSetsIn (C i)) {μν : Measure (∀ i, α i)}
    (h₁ : ∀ s : ∀ i, Set (α i), (∀ i, s i ∈ C i) → μν (pi univ s) = ∏ i, μ i (s i)) :
    Measure.pi μ = μν 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} {C : (i : ι) → Set (Set (α i))} (hC : ∀ (i : ι), MeasurableSpace.generateFrom (C i) = inst✝ i) (h2C : ∀ (i : ι), IsPiSystem (C i)) (h3C : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)) {μν : MeasureTheory.Measure ((i : ι) → α i)} (h₁ : ∀ (s : (i : ι) → Set (α i)), (∀ (i : ι), s i ∈ C i) → μν (Set.univ.pi s) = ∏ i : ι, (μ i) (s i))
Goal: MeasureTheory.Measure.pi μ = μν
-- MeasureTheory.Measure.pi_eq
theorem pi_eq {μ' : Measure (∀ i, α i)}
    (h : ∀ s : ∀ i, Set (α i), (∀ i, MeasurableSet (s i)) → μ' (pi univ s) = ∏ i, μ i (s i)) :
    Measure.pi μ = μ' 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] {μ' : MeasureTheory.Measure ((i : ι) → α i)} (h : ∀ (s : (i : ι) → Set (α i)), (∀ (i : ι), MeasurableSet (s i)) → μ' (Set.univ.pi s) = ∏ i : ι, (μ i) (s i))
Goal: MeasureTheory.Measure.pi μ = μ'
-- MeasureTheory.Measure.pi'_eq_pi
theorem pi'_eq_pi [Encodable ι] : pi' μ = Measure.pi μ 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [Encodable ι]
Goal: MeasureTheory.Measure.pi' μ = MeasureTheory.Measure.pi μ
-- MeasureTheory.Measure.pi_pi
@[simp]
theorem pi_pi (s : ∀ i, Set (α i)) : Measure.pi μ (pi univ s) = ∏ i, μ i (s i) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] (s : (i : ι) → Set (α i))
Goal: (MeasureTheory.Measure.pi μ) (Set.univ.pi s) = ∏ i : ι, (μ i) (s i)
-- MeasureTheory.Measure.pi_univ
nonrec theorem pi_univ : Measure.pi μ univ = ∏ i, μ i univ 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)]
Goal: (MeasureTheory.Measure.pi μ) Set.univ = ∏ i : ι, (μ i) Set.univ
-- MeasureTheory.Measure.pi_ball
theorem pi_ball [∀ i, MetricSpace (α i)] (x : ∀ i, α i) {r : ℝ} (hr : 0 < r) :
    Measure.pi μ (Metric.ball x r) = ∏ i, μ i (Metric.ball (x i) r) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → MetricSpace (α i)] (x : (i : ι) → α i) {r : ℝ} (hr : 0 < r)
Goal: (MeasureTheory.Measure.pi μ) (Metric.ball x r) = ∏ i : ι, (μ i) (Metric.ball (x i) r)
-- MeasureTheory.Measure.pi_closedBall
theorem pi_closedBall [∀ i, MetricSpace (α i)] (x : ∀ i, α i) {r : ℝ} (hr : 0 ≤ r) :
    Measure.pi μ (Metric.closedBall x r) = ∏ i, μ i (Metric.closedBall (x i) r) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → MetricSpace (α i)] (x : (i : ι) → α i) {r : ℝ} (hr : 0 ≤ r)
Goal: (MeasureTheory.Measure.pi μ) (Metric.closedBall x r) = ∏ i : ι, (μ i) (Metric.closedBall (x i) r)
-- MeasureTheory.Measure.pi_of_empty
theorem pi_of_empty {α : Type*} [Fintype α] [IsEmpty α] {β : α → Type*}
    {m : ∀ a, MeasurableSpace (β a)} (μ : ∀ a : α, Measure (β a)) (x : ∀ a, β a := isEmptyElim) :
    Measure.pi μ = dirac x 
Arguments: {α : Type u_4} [Fintype α] [IsEmpty α] {β : α → Type u_5} {m : (a : α) → MeasurableSpace (β a)} (μ : (a : α) → MeasureTheory.Measure (β a)) (x : (a : α) → β a := fun a => isEmptyElim a)
Goal: MeasureTheory.Measure.pi μ = MeasureTheory.Measure.dirac x
-- MeasureTheory.Measure.volume_pi_eq_dirac
lemma volume_pi_eq_dirac {ι : Type*} [Fintype ι] [IsEmpty ι]
    {α : ι → Type*} [∀ i, MeasureSpace (α i)] (x : ∀ a, α a := isEmptyElim) :
    (volume : Measure (∀ i, α i)) = Measure.dirac x 
Arguments: {ι : Type u_4} [Fintype ι] [IsEmpty ι] {α : ι → Type u_5} [(i : ι) → MeasureTheory.MeasureSpace (α i)] (x : (a : ι) → α a := fun a => isEmptyElim a)
Goal: MeasureTheory.volume = MeasureTheory.Measure.dirac x
-- MeasureTheory.Measure.pi_empty_univ
@[simp]
theorem pi_empty_univ {α : Type*} [Fintype α] [IsEmpty α] {β : α → Type*}
    {m : ∀ α, MeasurableSpace (β α)} (μ : ∀ a : α, Measure (β a)) :
    Measure.pi μ (Set.univ) = 1 
Arguments: {α : Type u_4} [Fintype α] [IsEmpty α] {β : α → Type u_5} {m : (α : α) → MeasurableSpace (β α)} (μ : (a : α) → MeasureTheory.Measure (β a))
Goal: (MeasureTheory.Measure.pi μ) Set.univ = 1
-- MeasureTheory.Measure.pi_eval_preimage_null
theorem pi_eval_preimage_null {i : ι} {s : Set (α i)} (hs : μ i s = 0) :
    Measure.pi μ (eval i ⁻¹' s) = 0 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] {i : ι} {s : Set (α i)} (hs : (μ i) s = 0)
Goal: (MeasureTheory.Measure.pi μ) (Function.eval i ⁻¹' s) = 0
-- MeasureTheory.Measure.pi_hyperplane
theorem pi_hyperplane (i : ι) [NoAtoms (μ i)] (x : α i) :
    Measure.pi μ { f : ∀ i, α i | f i = x } = 0 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] (i : ι) [MeasureTheory.NoAtoms (μ i)] (x : α i)
Goal: (MeasureTheory.Measure.pi μ) {f | f i = x} = 0
-- MeasureTheory.Measure.ae_eval_ne
theorem ae_eval_ne (i : ι) [NoAtoms (μ i)] (x : α i) : ∀ᵐ y : ∀ i, α i ∂Measure.pi μ, y i ≠ x 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] (i : ι) [MeasureTheory.NoAtoms (μ i)] (x : α i)
Goal: ∀ᵐ (y : (i : ι) → α i) ∂MeasureTheory.Measure.pi μ, y i ≠ x
-- MeasureTheory.Measure.tendsto_eval_ae_ae
theorem tendsto_eval_ae_ae {i : ι} : Tendsto (eval i) (ae (Measure.pi μ)) (ae (μ i)) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] {i : ι}
Goal: Filter.Tendsto (Function.eval i) (MeasureTheory.ae (MeasureTheory.Measure.pi μ)) (MeasureTheory.ae (μ i))
-- MeasureTheory.Measure.ae_pi_le_pi
theorem ae_pi_le_pi : ae (Measure.pi μ) ≤ Filter.pi fun i => ae (μ i) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)]
Goal: MeasureTheory.ae (MeasureTheory.Measure.pi μ) ≤ Filter.pi fun i => MeasureTheory.ae (μ i)
-- MeasureTheory.Measure.ae_eq_pi
theorem ae_eq_pi {β : ι → Type*} {f f' : ∀ i, α i → β i} (h : ∀ i, f i =ᵐ[μ i] f' i) :
    (fun (x : ∀ i, α i) i => f i (x i)) =ᵐ[Measure.pi μ] fun x i => f' i (x i) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] {β : ι → Type u_4} {f f' : (i : ι) → α i → β i} (h : ∀ (i : ι), f i =ᵐ[μ i] f' i)
Goal: (fun x i => f i (x i)) =ᵐ[MeasureTheory.Measure.pi μ] fun x i => f' i (x i)
-- MeasureTheory.Measure.ae_le_pi
theorem ae_le_pi {β : ι → Type*} [∀ i, Preorder (β i)] {f f' : ∀ i, α i → β i}
    (h : ∀ i, f i ≤ᵐ[μ i] f' i) :
    (fun (x : ∀ i, α i) i => f i (x i)) ≤ᵐ[Measure.pi μ] fun x i => f' i (x i) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] {β : ι → Type u_4} [(i : ι) → Preorder (β i)] {f f' : (i : ι) → α i → β i} (h : ∀ (i : ι), f i ≤ᵐ[μ i] f' i)
Goal: (fun x i => f i (x i)) ≤ᵐ[MeasureTheory.Measure.pi μ] fun x i => f' i (x i)
-- MeasureTheory.Measure.ae_le_set_pi
theorem ae_le_set_pi {I : Set ι} {s t : ∀ i, Set (α i)} (h : ∀ i ∈ I, s i ≤ᵐ[μ i] t i) :
    Set.pi I s ≤ᵐ[Measure.pi μ] Set.pi I t 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] {I : Set ι} {s t : (i : ι) → Set (α i)} (h : ∀ i ∈ I, s i ≤ᵐ[μ i] t i)
Goal: I.pi s ≤ᵐ[MeasureTheory.Measure.pi μ] I.pi t
-- MeasureTheory.Measure.ae_eq_set_pi
theorem ae_eq_set_pi {I : Set ι} {s t : ∀ i, Set (α i)} (h : ∀ i ∈ I, s i =ᵐ[μ i] t i) :
    Set.pi I s =ᵐ[Measure.pi μ] Set.pi I t 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] {I : Set ι} {s t : (i : ι) → Set (α i)} (h : ∀ i ∈ I, s i =ᵐ[μ i] t i)
Goal: I.pi s =ᵐ[MeasureTheory.Measure.pi μ] I.pi t
-- MeasureTheory.Measure.pi_map_piCongrLeft
lemma pi_map_piCongrLeft [hι' : Fintype ι'] (e : ι ≃ ι') {β : ι' → Type*}
    [∀ i, MeasurableSpace (β i)] (μ : (i : ι') → Measure (β i)) [∀ i, SigmaFinite (μ i)] :
    (Measure.pi fun i ↦ μ (e i)).map (MeasurableEquiv.piCongrLeft (fun i ↦ β i) e)
      = Measure.pi μ 
Arguments: {ι : Type u_1} {ι' : Type u_2} [Fintype ι] [hι' : Fintype ι'] (e : ι ≃ ι') {β : ι' → Type u_4} [(i : ι') → MeasurableSpace (β i)] (μ : (i : ι') → MeasureTheory.Measure (β i)) [∀ (i : ι'), MeasureTheory.SigmaFinite (μ i)]
Goal: MeasureTheory.Measure.map (⇑(MeasurableEquiv.piCongrLeft (fun i => β i) e))
    (MeasureTheory.Measure.pi fun i => μ (e i)) =
  MeasureTheory.Measure.pi μ
-- MeasureTheory.Measure.pi_map_piOptionEquivProd
lemma pi_map_piOptionEquivProd {β : Option ι → Type*} [∀ i, MeasurableSpace (β i)]
    (μ : (i : Option ι) → Measure (β i)) [∀ (i : Option ι), SigmaFinite (μ i)] :
    ((Measure.pi fun i ↦ μ (some i)).prod (μ none)).map
      (MeasurableEquiv.piOptionEquivProd β).symm = Measure.pi μ 
Arguments: {ι : Type u_1} [Fintype ι] {β : Option ι → Type u_4} [(i : Option ι) → MeasurableSpace (β i)] (μ : (i : Option ι) → MeasureTheory.Measure (β i)) [∀ (i : Option ι), MeasureTheory.SigmaFinite (μ i)]
Goal: MeasureTheory.Measure.map (⇑(MeasurableEquiv.piOptionEquivProd β).symm)
    ((MeasureTheory.Measure.pi fun i => μ (some i)).prod (μ none)) =
  MeasureTheory.Measure.pi μ
-- MeasureTheory.Measure.pi_Iio_ae_eq_pi_Iic
theorem pi_Iio_ae_eq_pi_Iic {s : Set ι} {f : ∀ i, α i} :
    (pi s fun i => Iio (f i)) =ᵐ[Measure.pi μ] pi s fun i => Iic (f i) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] {s : Set ι} {f : (i : ι) → α i}
Goal: (s.pi fun i => Set.Iio (f i)) =ᵐ[MeasureTheory.Measure.pi μ] s.pi fun i => Set.Iic (f i)
-- MeasureTheory.Measure.pi_Ioi_ae_eq_pi_Ici
theorem pi_Ioi_ae_eq_pi_Ici {s : Set ι} {f : ∀ i, α i} :
    (pi s fun i => Ioi (f i)) =ᵐ[Measure.pi μ] pi s fun i => Ici (f i) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] {s : Set ι} {f : (i : ι) → α i}
Goal: (s.pi fun i => Set.Ioi (f i)) =ᵐ[MeasureTheory.Measure.pi μ] s.pi fun i => Set.Ici (f i)
-- MeasureTheory.Measure.univ_pi_Iio_ae_eq_Iic
theorem univ_pi_Iio_ae_eq_Iic {f : ∀ i, α i} :
    (pi univ fun i => Iio (f i)) =ᵐ[Measure.pi μ] Iic f 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] {f : (i : ι) → α i}
Goal: (Set.univ.pi fun i => Set.Iio (f i)) =ᵐ[MeasureTheory.Measure.pi μ] Set.Iic f
-- MeasureTheory.Measure.univ_pi_Ioi_ae_eq_Ici
theorem univ_pi_Ioi_ae_eq_Ici {f : ∀ i, α i} :
    (pi univ fun i => Ioi (f i)) =ᵐ[Measure.pi μ] Ici f 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] {f : (i : ι) → α i}
Goal: (Set.univ.pi fun i => Set.Ioi (f i)) =ᵐ[MeasureTheory.Measure.pi μ] Set.Ici f
-- MeasureTheory.Measure.pi_Ioo_ae_eq_pi_Icc
theorem pi_Ioo_ae_eq_pi_Icc {s : Set ι} {f g : ∀ i, α i} :
    (pi s fun i => Ioo (f i) (g i)) =ᵐ[Measure.pi μ] pi s fun i => Icc (f i) (g i) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] {s : Set ι} {f g : (i : ι) → α i}
Goal: (s.pi fun i => Set.Ioo (f i) (g i)) =ᵐ[MeasureTheory.Measure.pi μ] s.pi fun i => Set.Icc (f i) (g i)
-- MeasureTheory.Measure.pi_Ioo_ae_eq_pi_Ioc
theorem pi_Ioo_ae_eq_pi_Ioc {s : Set ι} {f g : ∀ i, α i} :
    (pi s fun i => Ioo (f i) (g i)) =ᵐ[Measure.pi μ] pi s fun i => Ioc (f i) (g i) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] {s : Set ι} {f g : (i : ι) → α i}
Goal: (s.pi fun i => Set.Ioo (f i) (g i)) =ᵐ[MeasureTheory.Measure.pi μ] s.pi fun i => Set.Ioc (f i) (g i)
-- MeasureTheory.Measure.univ_pi_Ioo_ae_eq_Icc
theorem univ_pi_Ioo_ae_eq_Icc {f g : ∀ i, α i} :
    (pi univ fun i => Ioo (f i) (g i)) =ᵐ[Measure.pi μ] Icc f g 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] {f g : (i : ι) → α i}
Goal: (Set.univ.pi fun i => Set.Ioo (f i) (g i)) =ᵐ[MeasureTheory.Measure.pi μ] Set.Icc f g
-- MeasureTheory.Measure.pi_Ioc_ae_eq_pi_Icc
theorem pi_Ioc_ae_eq_pi_Icc {s : Set ι} {f g : ∀ i, α i} :
    (pi s fun i => Ioc (f i) (g i)) =ᵐ[Measure.pi μ] pi s fun i => Icc (f i) (g i) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] {s : Set ι} {f g : (i : ι) → α i}
Goal: (s.pi fun i => Set.Ioc (f i) (g i)) =ᵐ[MeasureTheory.Measure.pi μ] s.pi fun i => Set.Icc (f i) (g i)
-- MeasureTheory.Measure.univ_pi_Ioc_ae_eq_Icc
theorem univ_pi_Ioc_ae_eq_Icc {f g : ∀ i, α i} :
    (pi univ fun i => Ioc (f i) (g i)) =ᵐ[Measure.pi μ] Icc f g 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] {f g : (i : ι) → α i}
Goal: (Set.univ.pi fun i => Set.Ioc (f i) (g i)) =ᵐ[MeasureTheory.Measure.pi μ] Set.Icc f g
-- MeasureTheory.Measure.pi_Ico_ae_eq_pi_Icc
theorem pi_Ico_ae_eq_pi_Icc {s : Set ι} {f g : ∀ i, α i} :
    (pi s fun i => Ico (f i) (g i)) =ᵐ[Measure.pi μ] pi s fun i => Icc (f i) (g i) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] {s : Set ι} {f g : (i : ι) → α i}
Goal: (s.pi fun i => Set.Ico (f i) (g i)) =ᵐ[MeasureTheory.Measure.pi μ] s.pi fun i => Set.Icc (f i) (g i)
-- MeasureTheory.Measure.univ_pi_Ico_ae_eq_Icc
theorem univ_pi_Ico_ae_eq_Icc {f g : ∀ i, α i} :
    (pi univ fun i => Ico (f i) (g i)) =ᵐ[Measure.pi μ] Icc f g 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] [(i : ι) → PartialOrder (α i)] [∀ (i : ι), MeasureTheory.NoAtoms (μ i)] {f g : (i : ι) → α i}
Goal: (Set.univ.pi fun i => Set.Ico (f i) (g i)) =ᵐ[MeasureTheory.Measure.pi μ] Set.Icc f g
-- MeasureTheory.Measure.pi_noAtoms
theorem pi_noAtoms (i : ι) [NoAtoms (μ i)] : NoAtoms (Measure.pi μ) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] {μ : (i : ι) → MeasureTheory.Measure (α i)} [∀ (i : ι), MeasureTheory.SigmaFinite (μ i)] (i : ι) [MeasureTheory.NoAtoms (μ i)]
Goal: MeasureTheory.NoAtoms (MeasureTheory.Measure.pi μ)
-- MeasureTheory.volume_pi
theorem volume_pi [∀ i, MeasureSpace (α i)] :
    (volume : Measure (∀ i, α i)) = Measure.pi fun _ => volume 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Fintype ι] [(i : ι) → MeasureTheory.MeasureSpace (α i)]
Goal: MeasureTheory.volume = MeasureTheory.Measure.pi fun x => MeasureTheory.volume