ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.extend_congr
theorem extend_congr {β : Type*} {Pb : β → Prop} {mb : ∀ s : β, Pb s → ℝ≥0∞} {sa : α} {sb : β}
    (hP : P sa ↔ Pb sb) (hm : ∀ (ha : P sa) (hb : Pb sb), m sa ha = mb sb hb) :
    extend m sa = extend mb sb 
Arguments: {α : Type u_1} {P : α → Prop} (m : (s : α) → P s → ℝ≥0∞) {β : Type u_2} {Pb : β → Prop} {mb : (s : β) → Pb s → ℝ≥0∞} {sa : α} {sb : β} (hP : P sa ↔ Pb sb) (hm : ∀ (ha : P sa) (hb : Pb sb), m sa ha = mb sb hb)
Goal: MeasureTheory.extend m sa = MeasureTheory.extend mb sb
-- MeasureTheory.extend_top
@[simp]
theorem extend_top {α : Type*} {P : α → Prop} : extend (fun _ _ => ∞ : ∀ s : α, P s → ℝ≥0∞) = ⊤ 
Arguments: {α : Type u_2} {P : α → Prop}
Goal: (MeasureTheory.extend fun x x => ⊤) = ⊤
-- MeasureTheory.extend_empty
theorem extend_empty : extend m ∅ = 0 
Arguments: {α : Type u_1} {P : Set α → Prop} {m : (s : Set α) → P s → ℝ≥0∞} (P0 : P ∅) (m0 : m ∅ P0 = 0)
Goal: MeasureTheory.extend m ∅ = 0
-- MeasureTheory.extend_iUnion_nat
theorem extend_iUnion_nat {f : ℕ → Set α} (hm : ∀ i, P (f i))
    (mU : m (⋃ i, f i) (PU hm) = ∑' i, m (f i) (hm i)) :
    extend m (⋃ i, f i) = ∑' i, extend m (f i) 
Arguments: {α : Type u_1} {P : Set α → Prop} {m : (s : Set α) → P s → ℝ≥0∞} (PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)) {f : ℕ → Set α} (hm : ∀ (i : ℕ), P (f i)) (mU : m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯)
Goal: MeasureTheory.extend m (⋃ i, f i) = ∑' (i : ℕ), MeasureTheory.extend m (f i)
-- MeasureTheory.extend_iUnion_le_tsum_nat'
theorem extend_iUnion_le_tsum_nat' (s : ℕ → Set α) :
    extend m (⋃ i, s i) ≤ ∑' i, extend m (s i) 
Arguments: {α : Type u_1} {P : Set α → Prop} {m : (s : Set α) → P s → ℝ≥0∞} (PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)) (msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯) (s : ℕ → Set α)
Goal: MeasureTheory.extend m (⋃ i, s i) ≤ ∑' (i : ℕ), MeasureTheory.extend m (s i)
-- MeasureTheory.extend_mono'
theorem extend_mono' ⦃s₁ s₂ : Set α⦄ (h₁ : P s₁) (hs : s₁ ⊆ s₂) : extend m s₁ ≤ extend m s₂ 
Arguments: {α : Type u_1} {P : Set α → Prop} {m : (s : Set α) → P s → ℝ≥0∞} (m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂) ⦃s₁ s₂ : Set α⦄ (h₁ : P s₁) (hs : s₁ ⊆ s₂)
Goal: MeasureTheory.extend m s₁ ≤ MeasureTheory.extend m s₂
-- MeasureTheory.extend_iUnion
theorem extend_iUnion {β} [Countable β] {f : β → Set α} (hd : Pairwise (Disjoint on f))
    (hm : ∀ i, P (f i)) : extend m (⋃ i, f i) = ∑' i, extend m (f i) 
Arguments: {α : Type u_1} {P : Set α → Prop} {m : (s : Set α) → P s → ℝ≥0∞} (P0 : P ∅) (m0 : m ∅ P0 = 0) (PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)) (mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯) {β : Type u_2} [Countable β] {f : β → Set α} (hd : Pairwise (Disjoint on f)) (hm : ∀ (i : β), P (f i))
Goal: MeasureTheory.extend m (⋃ i, f i) = ∑' (i : β), MeasureTheory.extend m (f i)
-- MeasureTheory.extend_union
theorem extend_union {s₁ s₂ : Set α} (hd : Disjoint s₁ s₂) (h₁ : P s₁) (h₂ : P s₂) :
    extend m (s₁ ∪ s₂) = extend m s₁ + extend m s₂ 
Arguments: {α : Type u_1} {P : Set α → Prop} {m : (s : Set α) → P s → ℝ≥0∞} (P0 : P ∅) (m0 : m ∅ P0 = 0) (PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)) (mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯) {s₁ s₂ : Set α} (hd : Disjoint s₁ s₂) (h₁ : P s₁) (h₂ : P s₂)
Goal: MeasureTheory.extend m (s₁ ∪ s₂) = MeasureTheory.extend m s₁ + MeasureTheory.extend m s₂
-- MeasureTheory.le_inducedOuterMeasure
theorem le_inducedOuterMeasure {μ : OuterMeasure α} :
    μ ≤ inducedOuterMeasure m P0 m0 ↔ ∀ (s) (hs : P s), μ s ≤ m s hs 
Arguments: {α : Type u_1} {P : Set α → Prop} {m : (s : Set α) → P s → ℝ≥0∞} {P0 : P ∅} {m0 : m ∅ P0 = 0} {μ : MeasureTheory.OuterMeasure α}
Goal: μ ≤ MeasureTheory.inducedOuterMeasure m P0 m0 ↔ ∀ (s : Set α) (hs : P s), μ s ≤ m s hs
-- MeasureTheory.inducedOuterMeasure_union_of_false_of_nonempty_inter
theorem inducedOuterMeasure_union_of_false_of_nonempty_inter {s t : Set α}
    (h : ∀ u, (s ∩ u).Nonempty → (t ∩ u).Nonempty → ¬P u) :
    inducedOuterMeasure m P0 m0 (s ∪ t) =
      inducedOuterMeasure m P0 m0 s + inducedOuterMeasure m P0 m0 t 
Arguments: {α : Type u_1} {P : Set α → Prop} {m : (s : Set α) → P s → ℝ≥0∞} {P0 : P ∅} {m0 : m ∅ P0 = 0} {s t : Set α} (h : ∀ (u : Set α), (s ∩ u).Nonempty → (t ∩ u).Nonempty → ¬P u)
Goal: (MeasureTheory.inducedOuterMeasure m P0 m0) (s ∪ t) =
  (MeasureTheory.inducedOuterMeasure m P0 m0) s + (MeasureTheory.inducedOuterMeasure m P0 m0) t
-- MeasureTheory.inducedOuterMeasure_eq_extend'
theorem inducedOuterMeasure_eq_extend' {s : Set α} (hs : P s) :
    inducedOuterMeasure m P0 m0 s = extend m s 
Arguments: {α : Type u_1} {P : Set α → Prop} {m : (s : Set α) → P s → ℝ≥0∞} {P0 : P ∅} {m0 : m ∅ P0 = 0} (PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)) (msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯) (m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂) {s : Set α} (hs : P s)
Goal: (MeasureTheory.inducedOuterMeasure m P0 m0) s = MeasureTheory.extend m s
-- MeasureTheory.inducedOuterMeasure_eq'
theorem inducedOuterMeasure_eq' {s : Set α} (hs : P s) : inducedOuterMeasure m P0 m0 s = m s hs 
Arguments: {α : Type u_1} {P : Set α → Prop} {m : (s : Set α) → P s → ℝ≥0∞} {P0 : P ∅} {m0 : m ∅ P0 = 0} (PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)) (msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯) (m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂) {s : Set α} (hs : P s)
Goal: (MeasureTheory.inducedOuterMeasure m P0 m0) s = m s hs
-- MeasureTheory.inducedOuterMeasure_eq_iInf
theorem inducedOuterMeasure_eq_iInf (s : Set α) :
    inducedOuterMeasure m P0 m0 s = ⨅ (t : Set α) (ht : P t) (_ : s ⊆ t), m t ht 
Arguments: {α : Type u_1} {P : Set α → Prop} {m : (s : Set α) → P s → ℝ≥0∞} {P0 : P ∅} {m0 : m ∅ P0 = 0} (PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)) (msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯) (m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂) (s : Set α)
Goal: (MeasureTheory.inducedOuterMeasure m P0 m0) s = ⨅ t, ⨅ (ht : P t), ⨅ (_ : s ⊆ t), m t ht
-- MeasureTheory.inducedOuterMeasure_preimage
theorem inducedOuterMeasure_preimage (f : α ≃ α) (Pm : ∀ s : Set α, P (f ⁻¹' s) ↔ P s)
    (mm : ∀ (s : Set α) (hs : P s), m (f ⁻¹' s) ((Pm _).mpr hs) = m s hs) {A : Set α} :
    inducedOuterMeasure m P0 m0 (f ⁻¹' A) = inducedOuterMeasure m P0 m0 A 
Arguments: {α : Type u_1} {P : Set α → Prop} {m : (s : Set α) → P s → ℝ≥0∞} {P0 : P ∅} {m0 : m ∅ P0 = 0} (PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)) (msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯) (m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂) (f : α ≃ α) (Pm : ∀ (s : Set α), P (⇑f ⁻¹' s) ↔ P s) (mm : ∀ (s : Set α) (hs : P s), m (⇑f ⁻¹' s) ⋯ = m s hs) {A : Set α}
Goal: (MeasureTheory.inducedOuterMeasure m P0 m0) (⇑f ⁻¹' A) = (MeasureTheory.inducedOuterMeasure m P0 m0) A
-- MeasureTheory.inducedOuterMeasure_exists_set
theorem inducedOuterMeasure_exists_set {s : Set α} (hs : inducedOuterMeasure m P0 m0 s ≠ ∞)
    {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∃ t : Set α,
      P t ∧ s ⊆ t ∧ inducedOuterMeasure m P0 m0 t ≤ inducedOuterMeasure m P0 m0 s + ε 
Arguments: {α : Type u_1} {P : Set α → Prop} {m : (s : Set α) → P s → ℝ≥0∞} {P0 : P ∅} {m0 : m ∅ P0 = 0} (PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)) (msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯) (m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂) {s : Set α} (hs : (MeasureTheory.inducedOuterMeasure m P0 m0) s ≠ ⊤) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ t, P t ∧ s ⊆ t ∧ (MeasureTheory.inducedOuterMeasure m P0 m0) t ≤ (MeasureTheory.inducedOuterMeasure m P0 m0) s + ε
-- MeasureTheory.inducedOuterMeasure_caratheodory
theorem inducedOuterMeasure_caratheodory (s : Set α) :
    MeasurableSet[(inducedOuterMeasure m P0 m0).caratheodory] s ↔
      ∀ t : Set α,
        P t →
          inducedOuterMeasure m P0 m0 (t ∩ s) + inducedOuterMeasure m P0 m0 (t \ s) ≤
            inducedOuterMeasure m P0 m0 t 
Arguments: {α : Type u_1} {P : Set α → Prop} {m : (s : Set α) → P s → ℝ≥0∞} {P0 : P ∅} {m0 : m ∅ P0 = 0} (PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)) (msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯) (m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂) (s : Set α)
Goal: MeasurableSet s ↔
  ∀ (t : Set α),
    P t →
      (MeasureTheory.inducedOuterMeasure m P0 m0) (t ∩ s) + (MeasureTheory.inducedOuterMeasure m P0 m0) (t \ s) ≤
        (MeasureTheory.inducedOuterMeasure m P0 m0) t
-- MeasureTheory.extend_mono
theorem extend_mono {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁) (hs : s₁ ⊆ s₂) :
    extend m s₁ ≤ extend m s₂ 
Arguments: {α : Type u_1} [MeasurableSpace α] {m : (s : Set α) → MeasurableSet s → ℝ≥0∞} (m0 : m ∅ ⋯ = 0) (mU :
  ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),
    Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯) {s₁ s₂ : Set α} (h₁ : MeasurableSet s₁) (hs : s₁ ⊆ s₂)
Goal: MeasureTheory.extend m s₁ ≤ MeasureTheory.extend m s₂
-- MeasureTheory.extend_iUnion_le_tsum_nat
theorem extend_iUnion_le_tsum_nat : ∀ s : ℕ → Set α,
    extend m (⋃ i, s i) ≤ ∑' i, extend m (s i) 
Arguments: {α : Type u_1} [MeasurableSpace α] {m : (s : Set α) → MeasurableSet s → ℝ≥0∞} (m0 : m ∅ ⋯ = 0) (mU :
  ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),
    Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯) (s : ℕ → Set α)
Goal: MeasureTheory.extend m (⋃ i, s i) ≤ ∑' (i : ℕ), MeasureTheory.extend m (s i)
-- MeasureTheory.inducedOuterMeasure_eq_extend
theorem inducedOuterMeasure_eq_extend {s : Set α} (hs : MeasurableSet s) :
    inducedOuterMeasure m MeasurableSet.empty m0 s = extend m s 
Arguments: {α : Type u_1} [MeasurableSpace α] {m : (s : Set α) → MeasurableSet s → ℝ≥0∞} (m0 : m ∅ ⋯ = 0) (mU :
  ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),
    Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯) {s : Set α} (hs : MeasurableSet s)
Goal: (MeasureTheory.inducedOuterMeasure m ⋯ m0) s = MeasureTheory.extend m s
-- MeasureTheory.inducedOuterMeasure_eq
theorem inducedOuterMeasure_eq {s : Set α} (hs : MeasurableSet s) :
    inducedOuterMeasure m MeasurableSet.empty m0 s = m s hs 
Arguments: {α : Type u_1} [MeasurableSpace α] {m : (s : Set α) → MeasurableSet s → ℝ≥0∞} (m0 : m ∅ ⋯ = 0) (mU :
  ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), MeasurableSet (f i)),
    Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯) {s : Set α} (hs : MeasurableSet s)
Goal: (MeasureTheory.inducedOuterMeasure m ⋯ m0) s = m s hs
-- MeasureTheory.OuterMeasure.le_trim_iff
theorem le_trim_iff {m₁ m₂ : OuterMeasure α} :
    m₁ ≤ m₂.trim ↔ ∀ s, MeasurableSet s → m₁ s ≤ m₂ s 
Arguments: {α : Type u_1} [MeasurableSpace α] {m₁ m₂ : MeasureTheory.OuterMeasure α}
Goal: m₁ ≤ m₂.trim ↔ ∀ (s : Set α), MeasurableSet s → m₁ s ≤ m₂ s
-- MeasureTheory.OuterMeasure.le_trim
theorem le_trim : m ≤ m.trim 
Arguments: {α : Type u_1} [MeasurableSpace α] (m : MeasureTheory.OuterMeasure α)
Goal: m ≤ m.trim
-- MeasureTheory.OuterMeasure.trim_eq
@[simp] -- Porting note: added `simp`
theorem trim_eq {s : Set α} (hs : MeasurableSet s) : m.trim s = m s 
Arguments: {α : Type u_1} [MeasurableSpace α] (m : MeasureTheory.OuterMeasure α) {s : Set α} (hs : MeasurableSet s)
Goal: m.trim s = m s
-- MeasureTheory.OuterMeasure.trim_congr
theorem trim_congr {m₁ m₂ : OuterMeasure α} (H : ∀ {s : Set α}, MeasurableSet s → m₁ s = m₂ s) :
    m₁.trim = m₂.trim 
Arguments: {α : Type u_1} [MeasurableSpace α] {m₁ m₂ : MeasureTheory.OuterMeasure α} (H : ∀ {s : Set α}, MeasurableSet s → m₁ s = m₂ s)
Goal: m₁.trim = m₂.trim
-- MeasureTheory.OuterMeasure.trim_mono
@[mono]
theorem trim_mono : Monotone (trim : OuterMeasure α → OuterMeasure α) 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: Monotone MeasureTheory.OuterMeasure.trim
-- MeasureTheory.OuterMeasure.trim_anti_measurableSpace
theorem trim_anti_measurableSpace (m : OuterMeasure α) {m0 m1 : MeasurableSpace α}
    (h : m0 ≤ m1) : @trim _ m1 m ≤ @trim _ m0 m 
Arguments: {α : Type u_1} (m : MeasureTheory.OuterMeasure α) {m0 m1 : MeasurableSpace α} (h : m0 ≤ m1)
Goal: m.trim ≤ m.trim
-- MeasureTheory.OuterMeasure.trim_le_trim_iff
theorem trim_le_trim_iff {m₁ m₂ : OuterMeasure α} :
    m₁.trim ≤ m₂.trim ↔ ∀ s, MeasurableSet s → m₁ s ≤ m₂ s 
Arguments: {α : Type u_1} [MeasurableSpace α] {m₁ m₂ : MeasureTheory.OuterMeasure α}
Goal: m₁.trim ≤ m₂.trim ↔ ∀ (s : Set α), MeasurableSet s → m₁ s ≤ m₂ s
-- MeasureTheory.OuterMeasure.trim_eq_trim_iff
theorem trim_eq_trim_iff {m₁ m₂ : OuterMeasure α} :
    m₁.trim = m₂.trim ↔ ∀ s, MeasurableSet s → m₁ s = m₂ s 
Arguments: {α : Type u_1} [MeasurableSpace α] {m₁ m₂ : MeasureTheory.OuterMeasure α}
Goal: m₁.trim = m₂.trim ↔ ∀ (s : Set α), MeasurableSet s → m₁ s = m₂ s
-- MeasureTheory.OuterMeasure.trim_eq_iInf
theorem trim_eq_iInf (s : Set α) : m.trim s = ⨅ (t) (_ : s ⊆ t) (_ : MeasurableSet t), m t 
Arguments: {α : Type u_1} [MeasurableSpace α] (m : MeasureTheory.OuterMeasure α) (s : Set α)
Goal: m.trim s = ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), m t
-- MeasureTheory.OuterMeasure.trim_eq_iInf'
theorem trim_eq_iInf' (s : Set α) : m.trim s = ⨅ t : { t // s ⊆ t ∧ MeasurableSet t }, m t 
Arguments: {α : Type u_1} [MeasurableSpace α] (m : MeasureTheory.OuterMeasure α) (s : Set α)
Goal: m.trim s = ⨅ t, m ↑t
-- MeasureTheory.OuterMeasure.trim_trim
theorem trim_trim (m : OuterMeasure α) : m.trim.trim = m.trim 
Arguments: {α : Type u_1} [MeasurableSpace α] (m : MeasureTheory.OuterMeasure α)
Goal: m.trim.trim = m.trim
-- MeasureTheory.OuterMeasure.trim_top
@[simp]
theorem trim_top : (⊤ : OuterMeasure α).trim = ⊤ 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: ⊤.trim = ⊤
-- MeasureTheory.OuterMeasure.trim_zero
@[simp]
theorem trim_zero : (0 : OuterMeasure α).trim = 0 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: MeasureTheory.OuterMeasure.trim 0 = 0
-- MeasureTheory.OuterMeasure.trim_sum_ge
theorem trim_sum_ge {ι} (m : ι → OuterMeasure α) : (sum fun i => (m i).trim) ≤ (sum m).trim 
Arguments: {α : Type u_1} [MeasurableSpace α] {ι : Type u_2} (m : ι → MeasureTheory.OuterMeasure α)
Goal: (MeasureTheory.OuterMeasure.sum fun i => (m i).trim) ≤ (MeasureTheory.OuterMeasure.sum m).trim
-- MeasureTheory.OuterMeasure.exists_measurable_superset_eq_trim
theorem exists_measurable_superset_eq_trim (m : OuterMeasure α) (s : Set α) :
    ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = m.trim s 
Arguments: {α : Type u_1} [MeasurableSpace α] (m : MeasureTheory.OuterMeasure α) (s : Set α)
Goal: ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = m.trim s
-- MeasureTheory.OuterMeasure.exists_measurable_superset_of_trim_eq_zero
theorem exists_measurable_superset_of_trim_eq_zero {m : OuterMeasure α} {s : Set α}
    (h : m.trim s = 0) : ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {m : MeasureTheory.OuterMeasure α} {s : Set α} (h : m.trim s = 0)
Goal: ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t = 0
-- MeasureTheory.OuterMeasure.exists_measurable_superset_forall_eq_trim
theorem exists_measurable_superset_forall_eq_trim {ι} [Countable ι] (μ : ι → OuterMeasure α)
    (s : Set α) : ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ i, μ i t = (μ i).trim s 
Arguments: {α : Type u_1} [MeasurableSpace α] {ι : Sort u_2} [Countable ι] (μ : ι → MeasureTheory.OuterMeasure α) (s : Set α)
Goal: ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ (i : ι), (μ i) t = (μ i).trim s
-- MeasureTheory.OuterMeasure.trim_binop
theorem trim_binop {m₁ m₂ m₃ : OuterMeasure α} {op : ℝ≥0∞ → ℝ≥0∞ → ℝ≥0∞}
    (h : ∀ s, m₁ s = op (m₂ s) (m₃ s)) (s : Set α) : m₁.trim s = op (m₂.trim s) (m₃.trim s) 
Arguments: {α : Type u_1} [MeasurableSpace α] {m₁ m₂ m₃ : MeasureTheory.OuterMeasure α} {op : ℝ≥0∞ → ℝ≥0∞ → ℝ≥0∞} (h : ∀ (s : Set α), m₁ s = op (m₂ s) (m₃ s)) (s : Set α)
Goal: m₁.trim s = op (m₂.trim s) (m₃.trim s)
-- MeasureTheory.OuterMeasure.trim_op
theorem trim_op {m₁ m₂ : OuterMeasure α} {op : ℝ≥0∞ → ℝ≥0∞} (h : ∀ s, m₁ s = op (m₂ s))
    (s : Set α) : m₁.trim s = op (m₂.trim s) 
Arguments: {α : Type u_1} [MeasurableSpace α] {m₁ m₂ : MeasureTheory.OuterMeasure α} {op : ℝ≥0∞ → ℝ≥0∞} (h : ∀ (s : Set α), m₁ s = op (m₂ s)) (s : Set α)
Goal: m₁.trim s = op (m₂.trim s)
-- MeasureTheory.OuterMeasure.trim_add
theorem trim_add (m₁ m₂ : OuterMeasure α) : (m₁ + m₂).trim = m₁.trim + m₂.trim 
Arguments: {α : Type u_1} [MeasurableSpace α] (m₁ m₂ : MeasureTheory.OuterMeasure α)
Goal: (m₁ + m₂).trim = m₁.trim + m₂.trim
-- MeasureTheory.OuterMeasure.trim_smul
theorem trim_smul {R : Type*} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (c : R)
    (m : OuterMeasure α) : (c • m).trim = c • m.trim 
Arguments: {α : Type u_1} [MeasurableSpace α] {R : Type u_2} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (c : R) (m : MeasureTheory.OuterMeasure α)
Goal: (c • m).trim = c • m.trim
-- MeasureTheory.OuterMeasure.trim_sup
theorem trim_sup (m₁ m₂ : OuterMeasure α) : (m₁ ⊔ m₂).trim = m₁.trim ⊔ m₂.trim 
Arguments: {α : Type u_1} [MeasurableSpace α] (m₁ m₂ : MeasureTheory.OuterMeasure α)
Goal: (m₁ ⊔ m₂).trim = m₁.trim ⊔ m₂.trim
-- MeasureTheory.OuterMeasure.trim_iSup
theorem trim_iSup {ι} [Countable ι] (μ : ι → OuterMeasure α) :
    trim (⨆ i, μ i) = ⨆ i, trim (μ i) 
Arguments: {α : Type u_1} [MeasurableSpace α] {ι : Sort u_2} [Countable ι] (μ : ι → MeasureTheory.OuterMeasure α)
Goal: (⨆ i, μ i).trim = ⨆ i, (μ i).trim
-- MeasureTheory.OuterMeasure.restrict_trim
theorem restrict_trim {μ : OuterMeasure α} {s : Set α} (hs : MeasurableSet s) :
    (restrict s μ).trim = restrict s μ.trim 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.OuterMeasure α} {s : Set α} (hs : MeasurableSet s)
Goal: ((MeasureTheory.OuterMeasure.restrict s) μ).trim = (MeasureTheory.OuterMeasure.restrict s) μ.trim
-- MeasurableEmbedding.measurableSet_image'
structure MeasurableEmbedding [MeasurableSpace α] [MeasurableSpace β] (f : α → β) : Prop where
    protected injective : Injective f
    protected measurable : Measurable f
    protected measurableSet_image' : ∀ ⦃s⦄, MeasurableSet s → MeasurableSet (f '' s)
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} (self : MeasurableEmbedding f) ⦃s : Set α⦄
Goal: MeasurableSet s → MeasurableSet (f '' s)
-- MeasurableEmbedding.injective
structure MeasurableEmbedding [MeasurableSpace α] [MeasurableSpace β] (f : α → β) : Prop where
    protected injective : Injective f
    protected measurable : Measurable f
    protected measurableSet_image' : ∀ ⦃s⦄, MeasurableSet s → MeasurableSet (f '' s)
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} (self : MeasurableEmbedding f)
Goal: Function.Injective f
-- MeasurableEmbedding.measurable
structure MeasurableEmbedding [MeasurableSpace α] [MeasurableSpace β] (f : α → β) : Prop where
    protected injective : Injective f
    protected measurable : Measurable f
    protected measurableSet_image' : ∀ ⦃s⦄, MeasurableSet s → MeasurableSet (f '' s)
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} (self : MeasurableEmbedding f)
Goal: Measurable f
-- MeasurableEmbedding.measurableSet_image
theorem measurableSet_image (hf : MeasurableEmbedding f) :
    MeasurableSet (f '' s) ↔ MeasurableSet s 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} {mα : MeasurableSpace α} [MeasurableSpace β] {f : α → β} (hf : MeasurableEmbedding f)
Goal: MeasurableSet (f '' s) ↔ MeasurableSet s
-- MeasurableEmbedding.id
theorem id : MeasurableEmbedding (id : α → α) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α}
Goal: MeasurableEmbedding id
-- MeasurableEmbedding.comp
theorem comp (hg : MeasurableEmbedding g) (hf : MeasurableEmbedding f) :
    MeasurableEmbedding (g ∘ f) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ] {f : α → β} {g : β → γ} (hg : MeasurableEmbedding g) (hf : MeasurableEmbedding f)
Goal: MeasurableEmbedding (g ∘ f)
-- MeasurableEmbedding.subtype_coe
theorem subtype_coe (hs : MeasurableSet s) : MeasurableEmbedding ((↑) : s → α) 
Arguments: {α : Type u_1} {s : Set α} {mα : MeasurableSpace α} (hs : MeasurableSet s)
Goal: MeasurableEmbedding Subtype.val
-- MeasurableEmbedding.measurableSet_range
theorem measurableSet_range (hf : MeasurableEmbedding f) : MeasurableSet (range f) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} [MeasurableSpace β] {f : α → β} (hf : MeasurableEmbedding f)
Goal: MeasurableSet (Set.range f)
-- MeasurableEmbedding.measurableSet_preimage
theorem measurableSet_preimage (hf : MeasurableEmbedding f) {s : Set β} :
    MeasurableSet (f ⁻¹' s) ↔ MeasurableSet (s ∩ range f) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} [MeasurableSpace β] {f : α → β} (hf : MeasurableEmbedding f) {s : Set β}
Goal: MeasurableSet (f ⁻¹' s) ↔ MeasurableSet (s ∩ Set.range f)
-- MeasurableEmbedding.measurable_rangeSplitting
theorem measurable_rangeSplitting (hf : MeasurableEmbedding f) :
    Measurable (rangeSplitting f) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} [MeasurableSpace β] {f : α → β} (hf : MeasurableEmbedding f)
Goal: Measurable (Set.rangeSplitting f)
-- MeasurableEmbedding.measurable_extend
theorem measurable_extend (hf : MeasurableEmbedding f) {g : α → γ} {g' : β → γ} (hg : Measurable g)
    (hg' : Measurable g') : Measurable (extend f g g') 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ] {f : α → β} (hf : MeasurableEmbedding f) {g : α → γ} {g' : β → γ} (hg : Measurable g) (hg' : Measurable g')
Goal: Measurable (Function.extend f g g')
-- MeasurableEmbedding.exists_measurable_extend
theorem exists_measurable_extend (hf : MeasurableEmbedding f) {g : α → γ} (hg : Measurable g)
    (hne : β → Nonempty γ) : ∃ g' : β → γ, Measurable g' ∧ g' ∘ f = g 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ] {f : α → β} (hf : MeasurableEmbedding f) {g : α → γ} (hg : Measurable g) (hne : β → Nonempty γ)
Goal: ∃ g', Measurable g' ∧ g' ∘ f = g
-- MeasurableEmbedding.measurable_comp_iff
theorem measurable_comp_iff (hg : MeasurableEmbedding g) : Measurable (g ∘ f) ↔ Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ] {f : α → β} {g : β → γ} (hg : MeasurableEmbedding g)
Goal: Measurable (g ∘ f) ↔ Measurable f
-- MeasurableSet.exists_measurable_proj
theorem MeasurableSet.exists_measurable_proj {_ : MeasurableSpace α}
    (hs : MeasurableSet s) (hne : s.Nonempty) : ∃ f : α → s, Measurable f ∧ ∀ x : s, f x = x 
Arguments: {α : Type u_1} {s : Set α}
Goal: ∀ {x : MeasurableSpace α}, MeasurableSet s → s.Nonempty → ∃ f, Measurable f ∧ ∀ (x : ↑s), f ↑x = x
-- MeasurableEquiv.measurable_invFun
structure MeasurableEquiv (α β : Type*) [MeasurableSpace α] [MeasurableSpace β] extends α ≃ β where
    measurable_toFun : Measurable toEquiv
    measurable_invFun : Measurable toEquiv.symm
Arguments: {α : Type u_6} {β : Type u_7} [MeasurableSpace α] [MeasurableSpace β] (self : α ≃ᵐ β)
Goal: Measurable ⇑self.symm
-- MeasurableEquiv.measurable_toFun
structure MeasurableEquiv (α β : Type*) [MeasurableSpace α] [MeasurableSpace β] extends α ≃ β where
    measurable_toFun : Measurable toEquiv
    measurable_invFun : Measurable toEquiv.symm
Arguments: {α : Type u_6} {β : Type u_7} [MeasurableSpace α] [MeasurableSpace β] (self : α ≃ᵐ β)
Goal: Measurable ⇑self.toEquiv
-- MeasurableEquiv.toEquiv_injective
theorem toEquiv_injective : Injective (toEquiv : α ≃ᵐ β → α ≃ β) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β]
Goal: Function.Injective MeasurableEquiv.toEquiv
-- MeasurableEquiv.coe_toEquiv
@[simp]
theorem coe_toEquiv (e : α ≃ᵐ β) : (e.toEquiv : α → β) = e 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (e : α ≃ᵐ β)
Goal: ⇑e.toEquiv = ⇑e