ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.SimpleFunc.FinMeasSupp.map₂
protected theorem map₂ [Zero δ] (hf : f.FinMeasSupp μ) {g : α →ₛ γ} (hg : g.FinMeasSupp μ)
    {op : β → γ → δ} (H : op 0 0 = 0) : ((pair f g).map (Function.uncurry op)).FinMeasSupp μ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {m : MeasurableSpace α} [Zero β] [Zero γ] {μ : MeasureTheory.Measure α} {f : MeasureTheory.SimpleFunc α β} [Zero δ] (hf : f.FinMeasSupp μ) {g : MeasureTheory.SimpleFunc α γ} (hg : g.FinMeasSupp μ) {op : β → γ → δ} (H : op 0 0 = 0)
Goal: (MeasureTheory.SimpleFunc.map (Function.uncurry op) (f.pair g)).FinMeasSupp μ
-- MeasureTheory.SimpleFunc.FinMeasSupp.add
protected theorem add {β} [AddMonoid β] {f g : α →ₛ β} (hf : f.FinMeasSupp μ)
    (hg : g.FinMeasSupp μ) : (f + g).FinMeasSupp μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_5} [AddMonoid β] {f g : MeasureTheory.SimpleFunc α β} (hf : f.FinMeasSupp μ) (hg : g.FinMeasSupp μ)
Goal: (f + g).FinMeasSupp μ
-- MeasureTheory.SimpleFunc.FinMeasSupp.mul
protected theorem mul {β} [MonoidWithZero β] {f g : α →ₛ β} (hf : f.FinMeasSupp μ)
    (hg : g.FinMeasSupp μ) : (f * g).FinMeasSupp μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_5} [MonoidWithZero β] {f g : MeasureTheory.SimpleFunc α β} (hf : f.FinMeasSupp μ) (hg : g.FinMeasSupp μ)
Goal: (f * g).FinMeasSupp μ
-- MeasureTheory.SimpleFunc.FinMeasSupp.lintegral_lt_top
theorem lintegral_lt_top {f : α →ₛ ℝ≥0∞} (hm : f.FinMeasSupp μ) (hf : ∀ᵐ a ∂μ, f a ≠ ∞) :
    f.lintegral μ < ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : MeasureTheory.SimpleFunc α ℝ≥0∞} (hm : f.FinMeasSupp μ) (hf : ∀ᵐ (a : α) ∂μ, ↑f a ≠ ⊤)
Goal: f.lintegral μ < ⊤
-- MeasureTheory.SimpleFunc.FinMeasSupp.of_lintegral_ne_top
theorem of_lintegral_ne_top {f : α →ₛ ℝ≥0∞} (h : f.lintegral μ ≠ ∞) : f.FinMeasSupp μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : MeasureTheory.SimpleFunc α ℝ≥0∞} (h : f.lintegral μ ≠ ⊤)
Goal: f.FinMeasSupp μ
-- MeasureTheory.SimpleFunc.FinMeasSupp.iff_lintegral_lt_top
theorem iff_lintegral_lt_top {f : α →ₛ ℝ≥0∞} (hf : ∀ᵐ a ∂μ, f a ≠ ∞) :
    f.FinMeasSupp μ ↔ f.lintegral μ < ∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : MeasureTheory.SimpleFunc α ℝ≥0∞} (hf : ∀ᵐ (a : α) ∂μ, ↑f a ≠ ⊤)
Goal: f.FinMeasSupp μ ↔ f.lintegral μ < ⊤
-- MeasureTheory.SimpleFunc.induction
@[elab_as_elim]
protected theorem induction {α γ} [MeasurableSpace α] [AddMonoid γ] {P : SimpleFunc α γ → Prop}
    (h_ind :
      ∀ (c) {s} (hs : MeasurableSet s),
        P (SimpleFunc.piecewise s hs (SimpleFunc.const _ c) (SimpleFunc.const _ 0)))
    (h_add : ∀ ⦃f g : SimpleFunc α γ⦄, Disjoint (support f) (support g) → P f → P g → P (f + g))
    (f : SimpleFunc α γ) : P f 
Arguments: {α : Type u_5} {γ : Type u_6} [MeasurableSpace α] [AddMonoid γ] {P : MeasureTheory.SimpleFunc α γ → Prop} (h_ind :
  ∀ (c : γ) {s : Set α} (hs : MeasurableSet s),
    P
      (MeasureTheory.SimpleFunc.piecewise s hs (MeasureTheory.SimpleFunc.const α c)
        (MeasureTheory.SimpleFunc.const α 0))) (h_add :
  ∀ ⦃f g : MeasureTheory.SimpleFunc α γ⦄, Disjoint (Function.support ↑f) (Function.support ↑g) → P f → P g → P (f + g)) (f : MeasureTheory.SimpleFunc α γ)
Goal: P f
-- Measurable.add_simpleFunc
theorem _root_.Measurable.add_simpleFunc
    {E : Type*} {_ : MeasurableSpace α} [MeasurableSpace E] [AddGroup E] [MeasurableAdd E]
    {g : α → E} (hg : Measurable g) (f : SimpleFunc α E) :
    Measurable (g + (f : α → E)) 
Arguments: {α : Type u_1} {E : Type u_5}
Goal: ∀ {x : MeasurableSpace α} [inst : MeasurableSpace E] [inst_1 : AddGroup E] [inst_2 : MeasurableAdd E] {g : α → E},
  Measurable g → ∀ (f : MeasureTheory.SimpleFunc α E), Measurable (g + ↑f)
-- Measurable.simpleFunc_add
theorem _root_.Measurable.simpleFunc_add
    {E : Type*} {_ : MeasurableSpace α} [MeasurableSpace E] [AddGroup E] [MeasurableAdd E]
    {g : α → E} (hg : Measurable g) (f : SimpleFunc α E) :
    Measurable ((f : α → E) + g) 
Arguments: {α : Type u_1} {E : Type u_5}
Goal: ∀ {x : MeasurableSpace α} [inst : MeasurableSpace E] [inst_1 : AddGroup E] [inst_2 : MeasurableAdd E] {g : α → E},
  Measurable g → ∀ (f : MeasureTheory.SimpleFunc α E), Measurable (↑f + g)
-- Measurable.ennreal_induction
@[elab_as_elim]
theorem Measurable.ennreal_induction {α} [MeasurableSpace α] {P : (α → ℝ≥0∞) → Prop}
    (h_ind : ∀ (c : ℝ≥0∞) ⦃s⦄, MeasurableSet s → P (Set.indicator s fun _ => c))
    (h_add :
      ∀ ⦃f g : α → ℝ≥0∞⦄,
        Disjoint (support f) (support g) → Measurable f → Measurable g → P f → P g → P (f + g))
    (h_iSup :
      ∀ ⦃f : ℕ → α → ℝ≥0∞⦄, (∀ n, Measurable (f n)) → Monotone f → (∀ n, P (f n)) →
        P fun x => ⨆ n, f n x)
    ⦃f : α → ℝ≥0∞⦄ (hf : Measurable f) : P f 
Arguments: {α : Type u_1} [MeasurableSpace α] {P : (α → ℝ≥0∞) → Prop} (h_ind : ∀ (c : ℝ≥0∞) ⦃s : Set α⦄, MeasurableSet s → P (s.indicator fun x => c)) (h_add :
  ∀ ⦃f g : α → ℝ≥0∞⦄,
    Disjoint (Function.support f) (Function.support g) → Measurable f → Measurable g → P f → P g → P (f + g)) (h_iSup :
  ∀ ⦃f : ℕ → α → ℝ≥0∞⦄, (∀ (n : ℕ), Measurable (f n)) → Monotone f → (∀ (n : ℕ), P (f n)) → P fun x => ⨆ n, f n x) ⦃f : α → ℝ≥0∞⦄ (hf : Measurable f)
Goal: P f
-- MeasurableSpace.measurableSet_invariants
theorem measurableSet_invariants {f : α → α} {s : Set α} :
    MeasurableSet[invariants f] s ↔ MeasurableSet s ∧ f ⁻¹' s = s 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → α} {s : Set α}
Goal: MeasurableSet s ↔ MeasurableSet s ∧ f ⁻¹' s = s
-- MeasurableSpace.invariants_id
@[simp]
theorem invariants_id : invariants (id : α → α) = ‹MeasurableSpace α› 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: MeasurableSpace.invariants id = inst✝
-- MeasurableSpace.invariants_le
theorem invariants_le (f : α → α) : invariants f ≤ ‹MeasurableSpace α› 
Arguments: {α : Type u_1} [MeasurableSpace α] (f : α → α)
Goal: MeasurableSpace.invariants f ≤ inst✝
-- MeasurableSpace.inf_le_invariants_comp
theorem inf_le_invariants_comp (f g : α → α) :
    invariants f ⊓ invariants g ≤ invariants (f ∘ g) 
Arguments: {α : Type u_1} [MeasurableSpace α] (f g : α → α)
Goal: MeasurableSpace.invariants f ⊓ MeasurableSpace.invariants g ≤ MeasurableSpace.invariants (f ∘ g)
-- MeasurableSpace.le_invariants_iterate
theorem le_invariants_iterate (f : α → α) (n : ℕ) :
    invariants f ≤ invariants (f^[n]) 
Arguments: {α : Type u_1} [MeasurableSpace α] (f : α → α) (n : ℕ)
Goal: MeasurableSpace.invariants f ≤ MeasurableSpace.invariants f^[n]
-- MeasurableSpace.measurable_invariants_dom
theorem measurable_invariants_dom {f : α → α} {g : α → β} :
    Measurable[invariants f] g ↔ Measurable g ∧ ∀ s, MeasurableSet s → (g ∘ f) ⁻¹' s = g ⁻¹' s 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] {f : α → α} {g : α → β}
Goal: Measurable g ↔ Measurable g ∧ ∀ (s : Set β), MeasurableSet s → g ∘ f ⁻¹' s = g ⁻¹' s
-- MeasurableSpace.measurable_invariants_of_semiconj
theorem measurable_invariants_of_semiconj {fa : α → α} {fb : β → β} {g : α → β} (hg : Measurable g)
    (hfg : Semiconj g fa fb) : @Measurable _ _ (invariants fa) (invariants fb) g 
Arguments: {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] {fa : α → α} {fb : β → β} {g : α → β} (hg : Measurable g) (hfg : Function.Semiconj g fa fb)
Goal: Measurable g
-- MeasureTheory.OuterMeasure.isCaratheodory_iff_le'
theorem isCaratheodory_iff_le' {s : Set α} :
    IsCaratheodory m s ↔ ∀ t, m (t ∩ s) + m (t \ s) ≤ m t 
Arguments: {α : Type u} (m : MeasureTheory.OuterMeasure α) {s : Set α}
Goal: m.IsCaratheodory s ↔ ∀ (t : Set α), m (t ∩ s) + m (t \ s) ≤ m t
-- MeasureTheory.OuterMeasure.isCaratheodory_empty
@[simp]
theorem isCaratheodory_empty : IsCaratheodory m ∅ 
Arguments: {α : Type u} (m : MeasureTheory.OuterMeasure α)
Goal: m.IsCaratheodory ∅
-- MeasureTheory.OuterMeasure.isCaratheodory_compl
theorem isCaratheodory_compl : IsCaratheodory m s₁ → IsCaratheodory m s₁ᶜ 
Arguments: {α : Type u} (m : MeasureTheory.OuterMeasure α) {s₁ : Set α}
Goal: m.IsCaratheodory s₁ → m.IsCaratheodory s₁ᶜ
-- MeasureTheory.OuterMeasure.isCaratheodory_compl_iff
@[simp]
theorem isCaratheodory_compl_iff : IsCaratheodory m sᶜ ↔ IsCaratheodory m s 
Arguments: {α : Type u} (m : MeasureTheory.OuterMeasure α) {s : Set α}
Goal: m.IsCaratheodory sᶜ ↔ m.IsCaratheodory s
-- MeasureTheory.OuterMeasure.isCaratheodory_union
theorem isCaratheodory_union (h₁ : IsCaratheodory m s₁) (h₂ : IsCaratheodory m s₂) :
    IsCaratheodory m (s₁ ∪ s₂) 
Arguments: {α : Type u} (m : MeasureTheory.OuterMeasure α) {s₁ s₂ : Set α} (h₁ : m.IsCaratheodory s₁) (h₂ : m.IsCaratheodory s₂)
Goal: m.IsCaratheodory (s₁ ∪ s₂)
-- MeasureTheory.OuterMeasure.measure_inter_union
theorem measure_inter_union (h : s₁ ∩ s₂ ⊆ ∅) (h₁ : IsCaratheodory m s₁) {t : Set α} :
    m (t ∩ (s₁ ∪ s₂)) = m (t ∩ s₁) + m (t ∩ s₂) 
Arguments: {α : Type u} (m : MeasureTheory.OuterMeasure α) {s₁ s₂ : Set α} (h : s₁ ∩ s₂ ⊆ ∅) (h₁ : m.IsCaratheodory s₁) {t : Set α}
Goal: m (t ∩ (s₁ ∪ s₂)) = m (t ∩ s₁) + m (t ∩ s₂)
-- MeasureTheory.OuterMeasure.isCaratheodory_iUnion_lt
theorem isCaratheodory_iUnion_lt {s : ℕ → Set α} :
    ∀ {n : ℕ}, (∀ i < n, IsCaratheodory m (s i)) → IsCaratheodory m (⋃ i < n, s i)
  
Arguments: {α : Type u} (m : MeasureTheory.OuterMeasure α) {s : ℕ → Set α} {n : ℕ}
Goal: (∀ i < n, m.IsCaratheodory (s i)) → m.IsCaratheodory (⋃ i, ⋃ (_ : i < n), s i)
-- MeasureTheory.OuterMeasure.isCaratheodory_inter
theorem isCaratheodory_inter (h₁ : IsCaratheodory m s₁) (h₂ : IsCaratheodory m s₂) :
    IsCaratheodory m (s₁ ∩ s₂) 
Arguments: {α : Type u} (m : MeasureTheory.OuterMeasure α) {s₁ s₂ : Set α} (h₁ : m.IsCaratheodory s₁) (h₂ : m.IsCaratheodory s₂)
Goal: m.IsCaratheodory (s₁ ∩ s₂)
-- MeasureTheory.OuterMeasure.isCaratheodory_sum
theorem isCaratheodory_sum {s : ℕ → Set α} (h : ∀ i, IsCaratheodory m (s i))
    (hd : Pairwise (Disjoint on s)) {t : Set α} :
    ∀ {n}, (∑ i ∈ Finset.range n, m (t ∩ s i)) = m (t ∩ ⋃ i < n, s i)
  
Arguments: {α : Type u} (m : MeasureTheory.OuterMeasure α) {s : ℕ → Set α} (h : ∀ (i : ℕ), m.IsCaratheodory (s i)) (hd : Pairwise (Disjoint on s)) {t : Set α} {n : ℕ}
Goal: ∑ i ∈ Finset.range n, m (t ∩ s i) = m (t ∩ ⋃ i, ⋃ (_ : i < n), s i)
-- MeasureTheory.OuterMeasure.isCaratheodory_iUnion_nat
theorem isCaratheodory_iUnion_nat {s : ℕ → Set α} (h : ∀ i, IsCaratheodory m (s i))
    (hd : Pairwise (Disjoint on s)) : IsCaratheodory m (⋃ i, s i) 
Arguments: {α : Type u} (m : MeasureTheory.OuterMeasure α) {s : ℕ → Set α} (h : ∀ (i : ℕ), m.IsCaratheodory (s i)) (hd : Pairwise (Disjoint on s))
Goal: m.IsCaratheodory (⋃ i, s i)
-- MeasureTheory.OuterMeasure.f_iUnion
theorem f_iUnion {s : ℕ → Set α} (h : ∀ i, IsCaratheodory m (s i)) (hd : Pairwise (Disjoint on s)) :
    m (⋃ i, s i) = ∑' i, m (s i) 
Arguments: {α : Type u} (m : MeasureTheory.OuterMeasure α) {s : ℕ → Set α} (h : ∀ (i : ℕ), m.IsCaratheodory (s i)) (hd : Pairwise (Disjoint on s))
Goal: m (⋃ i, s i) = ∑' (i : ℕ), m (s i)
-- MeasureTheory.OuterMeasure.isCaratheodory_iff
theorem isCaratheodory_iff {s : Set α} :
    MeasurableSet[OuterMeasure.caratheodory m] s ↔ ∀ t, m t = m (t ∩ s) + m (t \ s) 
Arguments: {α : Type u} (m : MeasureTheory.OuterMeasure α) {s : Set α}
Goal: MeasurableSet s ↔ ∀ (t : Set α), m t = m (t ∩ s) + m (t \ s)
-- MeasureTheory.OuterMeasure.isCaratheodory_iff_le
theorem isCaratheodory_iff_le {s : Set α} :
    MeasurableSet[OuterMeasure.caratheodory m] s ↔ ∀ t, m (t ∩ s) + m (t \ s) ≤ m t 
Arguments: {α : Type u} (m : MeasureTheory.OuterMeasure α) {s : Set α}
Goal: MeasurableSet s ↔ ∀ (t : Set α), m (t ∩ s) + m (t \ s) ≤ m t
-- MeasureTheory.OuterMeasure.iUnion_eq_of_caratheodory
protected theorem iUnion_eq_of_caratheodory {s : ℕ → Set α}
    (h : ∀ i, MeasurableSet[OuterMeasure.caratheodory m] (s i)) (hd : Pairwise (Disjoint on s)) :
    m (⋃ i, s i) = ∑' i, m (s i) 
Arguments: {α : Type u} (m : MeasureTheory.OuterMeasure α) {s : ℕ → Set α} (h : ∀ (i : ℕ), MeasurableSet (s i)) (hd : Pairwise (Disjoint on s))
Goal: m (⋃ i, s i) = ∑' (i : ℕ), m (s i)
-- MeasureTheory.OuterMeasure.ofFunction_caratheodory
theorem ofFunction_caratheodory {m : Set α → ℝ≥0∞} {s : Set α} {h₀ : m ∅ = 0}
    (hs : ∀ t, m (t ∩ s) + m (t \ s) ≤ m t) :
    MeasurableSet[(OuterMeasure.ofFunction m h₀).caratheodory] s 
Arguments: {α : Type u_1} {m : Set α → ℝ≥0∞} {s : Set α} {h₀ : m ∅ = 0} (hs : ∀ (t : Set α), m (t ∩ s) + m (t \ s) ≤ m t)
Goal: MeasurableSet s
-- MeasureTheory.OuterMeasure.boundedBy_caratheodory
theorem boundedBy_caratheodory {m : Set α → ℝ≥0∞} {s : Set α}
    (hs : ∀ t, m (t ∩ s) + m (t \ s) ≤ m t) : MeasurableSet[(boundedBy m).caratheodory] s 
Arguments: {α : Type u_1} {m : Set α → ℝ≥0∞} {s : Set α} (hs : ∀ (t : Set α), m (t ∩ s) + m (t \ s) ≤ m t)
Goal: MeasurableSet s
-- MeasureTheory.OuterMeasure.zero_caratheodory
@[simp]
theorem zero_caratheodory : (0 : OuterMeasure α).caratheodory = ⊤ 
Arguments: {α : Type u_1}
Goal: MeasureTheory.OuterMeasure.caratheodory 0 = ⊤
-- MeasureTheory.OuterMeasure.top_caratheodory
theorem top_caratheodory : (⊤ : OuterMeasure α).caratheodory = ⊤ 
Arguments: {α : Type u_1}
Goal: ⊤.caratheodory = ⊤
-- MeasureTheory.OuterMeasure.le_add_caratheodory
theorem le_add_caratheodory (m₁ m₂ : OuterMeasure α) :
    m₁.caratheodory ⊓ m₂.caratheodory ≤ (m₁ + m₂ : OuterMeasure α).caratheodory 
Arguments: {α : Type u_1} (m₁ m₂ : MeasureTheory.OuterMeasure α)
Goal: m₁.caratheodory ⊓ m₂.caratheodory ≤ (m₁ + m₂).caratheodory
-- MeasureTheory.OuterMeasure.le_sum_caratheodory
theorem le_sum_caratheodory {ι} (m : ι → OuterMeasure α) :
    ⨅ i, (m i).caratheodory ≤ (sum m).caratheodory 
Arguments: {α : Type u_1} {ι : Type u_2} (m : ι → MeasureTheory.OuterMeasure α)
Goal: ⨅ i, (m i).caratheodory ≤ (MeasureTheory.OuterMeasure.sum m).caratheodory
-- MeasureTheory.OuterMeasure.le_smul_caratheodory
theorem le_smul_caratheodory (a : ℝ≥0∞) (m : OuterMeasure α) :
    m.caratheodory ≤ (a • m).caratheodory 
Arguments: {α : Type u_1} (a : ℝ≥0∞) (m : MeasureTheory.OuterMeasure α)
Goal: m.caratheodory ≤ (a • m).caratheodory
-- MeasureTheory.OuterMeasure.dirac_caratheodory
@[simp]
theorem dirac_caratheodory (a : α) : (dirac a).caratheodory = ⊤ 
Arguments: {α : Type u_1} (a : α)
Goal: (MeasureTheory.OuterMeasure.dirac a).caratheodory = ⊤
-- MeasureTheory.condexp_def
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
    (μ : Measure α) (f : α → F') : α → F' 
Arguments: {α : Type u_5} {F' : Type u_6} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] (m : MeasurableSpace α) {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → F')
Goal: μ[f|m] =
  if hm : m ≤ m0 then
    if h : MeasureTheory.SigmaFinite (μ.trim hm) ∧ MeasureTheory.Integrable f μ then
      if MeasureTheory.StronglyMeasurable f then f
      else MeasureTheory.AEStronglyMeasurable'.mk ↑↑(MeasureTheory.condexpL1 hm μ f) ⋯
    else 0
  else 0
-- MeasureTheory.condexp_of_not_le
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → F'} (hm_not : ¬m ≤ m0)
Goal: μ[f|m] = 0
-- MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
    μ[f|m] = 0 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → F'} (hm : m ≤ m0) (hμm_not : ¬MeasureTheory.SigmaFinite (μ.trim hm))
Goal: μ[f|m] = 0
-- MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
    μ[f|m] =
      if Integrable f μ then
        if StronglyMeasurable[m] f then f
        else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
      else 0 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → F'} (hm : m ≤ m0) [hμm : MeasureTheory.SigmaFinite (μ.trim hm)]
Goal: μ[f|m] =
  if MeasureTheory.Integrable f μ then
    if MeasureTheory.StronglyMeasurable f then f
    else MeasureTheory.AEStronglyMeasurable'.mk ↑↑(MeasureTheory.condexpL1 hm μ f) ⋯
  else 0
-- MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
    (hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) [hμm : MeasureTheory.SigmaFinite (μ.trim hm)] {f : α → F'} (hf : MeasureTheory.StronglyMeasurable f) (hfi : MeasureTheory.Integrable f μ)
Goal: μ[f|m] = f
-- MeasureTheory.condexp_const
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
    μ[fun _ : α => c|m] = fun _ => c 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) (c : F') [MeasureTheory.IsFiniteMeasure μ]
Goal: μ[fun x => c|m] = fun x => c
-- MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
    μ[f|m] =ᵐ[μ] condexpL1 hm μ f 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) [hμm : MeasureTheory.SigmaFinite (μ.trim hm)] (f : α → F')
Goal: μ[f|m] =ᵐ[μ] ↑↑(MeasureTheory.condexpL1 hm μ f)
-- MeasureTheory.condexp_ae_eq_condexpL1CLM
theorem condexp_ae_eq_condexpL1CLM (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
    μ[f|m] =ᵐ[μ] condexpL1CLM F' hm μ (hf.toL1 f) 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → F'} (hm : m ≤ m0) [MeasureTheory.SigmaFinite (μ.trim hm)] (hf : MeasureTheory.Integrable f μ)
Goal: μ[f|m] =ᵐ[μ] ↑↑((MeasureTheory.condexpL1CLM F' hm μ) (MeasureTheory.Integrable.toL1 f hf))
-- MeasureTheory.condexp_undef
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → F'} (hf : ¬MeasureTheory.Integrable f μ)
Goal: μ[f|m] = 0
-- MeasureTheory.condexp_zero
@[simp]
theorem condexp_zero : μ[(0 : α → F')|m] = 0 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: μ[0|m] = 0
-- MeasureTheory.stronglyMeasurable_condexp
theorem stronglyMeasurable_condexp : StronglyMeasurable[m] (μ[f|m]) 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → F'}
Goal: MeasureTheory.StronglyMeasurable (μ[f|m])
-- MeasureTheory.condexp_congr_ae
theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → F'} (h : f =ᵐ[μ] g)
Goal: μ[f|m] =ᵐ[μ] μ[g|m]
-- MeasureTheory.condexp_of_aestronglyMeasurable'
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
    (hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) [hμm : MeasureTheory.SigmaFinite (μ.trim hm)] {f : α → F'} (hf : MeasureTheory.AEStronglyMeasurable' m f μ) (hfi : MeasureTheory.Integrable f μ)
Goal: μ[f|m] =ᵐ[μ] f
-- MeasureTheory.integrable_condexp
theorem integrable_condexp : Integrable (μ[f|m]) μ 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → F'}
Goal: MeasureTheory.Integrable (μ[f|m]) μ
-- MeasureTheory.setIntegral_condexp
theorem setIntegral_condexp (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
    (hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → F'} {s : Set α} (hm : m ≤ m0) [MeasureTheory.SigmaFinite (μ.trim hm)] (hf : MeasureTheory.Integrable f μ) (hs : MeasurableSet s)
Goal: ∫ (x : α) in s, (μ[f|m]) x ∂μ = ∫ (x : α) in s, f x ∂μ
-- MeasureTheory.integral_condexp
theorem integral_condexp (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
    ∫ x, (μ[f|m]) x ∂μ = ∫ x, f x ∂μ 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → F'} (hm : m ≤ m0) [hμm : MeasureTheory.SigmaFinite (μ.trim hm)]
Goal: ∫ (x : α), (μ[f|m]) x ∂μ = ∫ (x : α), f x ∂μ
-- MeasureTheory.integral_condexp_indicator
theorem integral_condexp_indicator [mF : MeasurableSpace F] {Y : α → F} (hY : Measurable Y)
    [SigmaFinite (μ.trim hY.comap_le)] {A : Set α} (hA : MeasurableSet A) :
    ∫ x, (μ[(A.indicator fun _ ↦ (1 : ℝ)) | mF.comap Y]) x ∂μ = (μ A).toReal 
Arguments: {α : Type u_1} {F : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [mF : MeasurableSpace F] {Y : α → F} (hY : Measurable Y) [MeasureTheory.SigmaFinite (μ.trim ⋯)] {A : Set α} (hA : MeasurableSet A)
Goal: ∫ (x : α), (μ[A.indicator fun x => 1|MeasurableSpace.comap Y mF]) x ∂μ = (μ A).toReal
-- MeasureTheory.ae_eq_condexp_of_forall_setIntegral_eq
theorem ae_eq_condexp_of_forall_setIntegral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
    {f g : α → F'} (hf : Integrable f μ)
    (hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
    (hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
    (hgm : AEStronglyMeasurable' m g μ) : g =ᵐ[μ] μ[f|m] 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) [MeasureTheory.SigmaFinite (μ.trim hm)] {f g : α → F'} (hf : MeasureTheory.Integrable f μ) (hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn g s μ) (hg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, f x ∂μ) (hgm : MeasureTheory.AEStronglyMeasurable' m g μ)
Goal: g =ᵐ[μ] μ[f|m]
-- MeasureTheory.condexp_bot'
theorem condexp_bot' [hμ : NeZero μ] (f : α → F') :
    μ[f|⊥] = fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [hμ : NeZero μ] (f : α → F')
Goal: μ[f|⊥] = fun x => (μ Set.univ).toReal⁻¹ • ∫ (x : α), f x ∂μ
-- MeasureTheory.condexp_bot_ae_eq
theorem condexp_bot_ae_eq (f : α → F') :
    μ[f|⊥] =ᵐ[μ] fun _ => (μ Set.univ).toReal⁻¹ • ∫ x, f x ∂μ 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → F')
Goal: μ[f|⊥] =ᵐ[μ] fun x => (μ Set.univ).toReal⁻¹ • ∫ (x : α), f x ∂μ
-- MeasureTheory.condexp_bot
theorem condexp_bot [IsProbabilityMeasure μ] (f : α → F') : μ[f|⊥] = fun _ => ∫ x, f x ∂μ 
Arguments: {α : Type u_1} {F' : Type u_3} [NormedAddCommGroup F'] [NormedSpace ℝ F'] [CompleteSpace F'] {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsProbabilityMeasure μ] (f : α → F')
Goal: μ[f|⊥] = fun x => ∫ (x : α), f x ∂μ