ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Measure.InnerRegularCompactLTTop.innerRegular
class InnerRegularCompactLTTop (μ : Measure α) : Prop where
  protected innerRegular : InnerRegularWRT μ IsCompact (fun s ↦ MeasurableSet s ∧ μ s ≠ ∞)
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [self : μ.InnerRegularCompactLTTop]
Goal: μ.InnerRegularWRT IsCompact fun s => MeasurableSet s ∧ μ s ≠ ⊤
-- Set.exists_isOpen_lt_of_lt
theorem _root_.Set.exists_isOpen_lt_of_lt [OuterRegular μ] (A : Set α) (r : ℝ≥0∞) (hr : μ A < r) :
    ∃ U, U ⊇ A ∧ IsOpen U ∧ μ U < r 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [μ.OuterRegular] (A : Set α) (r : ℝ≥0∞) (hr : μ A < r)
Goal: ∃ U ⊇ A, IsOpen U ∧ μ U < r
-- Set.measure_eq_iInf_isOpen
theorem _root_.Set.measure_eq_iInf_isOpen (A : Set α) (μ : Measure α) [OuterRegular μ] :
    μ A = ⨅ (U : Set α) (_ : A ⊆ U) (_ : IsOpen U), μ U 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] (A : Set α) (μ : MeasureTheory.Measure α) [μ.OuterRegular]
Goal: μ A = ⨅ U, ⨅ (_ : A ⊆ U), ⨅ (_ : IsOpen U), μ U
-- Set.exists_isOpen_lt_add
theorem _root_.Set.exists_isOpen_lt_add [OuterRegular μ] (A : Set α) (hA : μ A ≠ ∞) {ε : ℝ≥0∞}
    (hε : ε ≠ 0) : ∃ U, U ⊇ A ∧ IsOpen U ∧ μ U < μ A + ε 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [μ.OuterRegular] (A : Set α) (hA : μ A ≠ ⊤) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ U ⊇ A, IsOpen U ∧ μ U < μ A + ε
-- Set.exists_isOpen_le_add
theorem _root_.Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ] {ε : ℝ≥0∞}
    (hε : ε ≠ 0) : ∃ U, U ⊇ A ∧ IsOpen U ∧ μ U ≤ μ A + ε 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] (A : Set α) (μ : MeasureTheory.Measure α) [μ.OuterRegular] {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ U ⊇ A, IsOpen U ∧ μ U ≤ μ A + ε
-- MeasurableSet.exists_isOpen_diff_lt
theorem _root_.MeasurableSet.exists_isOpen_diff_lt [OuterRegular μ] {A : Set α}
    (hA : MeasurableSet A) (hA' : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∃ U, U ⊇ A ∧ IsOpen U ∧ μ U < ∞ ∧ μ (U \ A) < ε 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [μ.OuterRegular] {A : Set α} (hA : MeasurableSet A) (hA' : μ A ≠ ⊤) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ U ⊇ A, IsOpen U ∧ μ U < ⊤ ∧ μ (U \ A) < ε
-- MeasureTheory.Measure.OuterRegular.map
protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace β]
    [BorelSpace β] (f : α ≃ₜ β) (μ : Measure α) [OuterRegular μ] :
    (Measure.map f μ).OuterRegular 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace β] [BorelSpace β] (f : α ≃ₜ β) (μ : MeasureTheory.Measure α) [μ.OuterRegular]
Goal: (MeasureTheory.Measure.map (⇑f) μ).OuterRegular
-- MeasureTheory.Measure.OuterRegular.smul
protected theorem smul (μ : Measure α) [OuterRegular μ] {x : ℝ≥0∞} (hx : x ≠ ∞) :
    (x • μ).OuterRegular 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] (μ : MeasureTheory.Measure α) [μ.OuterRegular] {x : ℝ≥0∞} (hx : x ≠ ⊤)
Goal: (x • μ).OuterRegular
-- MeasureTheory.Measure.OuterRegular.of_restrict
lemma of_restrict [OpensMeasurableSpace α] {μ : Measure α} {s : ℕ → Set α}
    (h : ∀ n, OuterRegular (μ.restrict (s n))) (h' : ∀ n, IsOpen (s n)) (h'' : univ ⊆ ⋃ n, s n) :
    OuterRegular μ 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] {μ : MeasureTheory.Measure α} {s : ℕ → Set α} (h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular) (h' : ∀ (n : ℕ), IsOpen (s n)) (h'' : Set.univ ⊆ ⋃ n, s n)
Goal: μ.OuterRegular
-- MeasureTheory.Measure.OuterRegular.measure_closure_eq_of_isCompact
lemma measure_closure_eq_of_isCompact [R1Space α] [OuterRegular μ]
    {k : Set α} (hk : IsCompact k) : μ (closure k) = μ k 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [R1Space α] [μ.OuterRegular] {k : Set α} (hk : IsCompact k)
Goal: μ (closure k) = μ k
-- MeasureTheory.Measure.FiniteSpanningSetsIn.outerRegular
protected theorem FiniteSpanningSetsIn.outerRegular [OpensMeasurableSpace α] {μ : Measure α}
    (s : μ.FiniteSpanningSetsIn { U | IsOpen U ∧ OuterRegular (μ.restrict U) }) :
    OuterRegular μ 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] {μ : MeasureTheory.Measure α} (s : μ.FiniteSpanningSetsIn {U | IsOpen U ∧ (μ.restrict U).OuterRegular})
Goal: μ.OuterRegular
-- MeasureTheory.Measure.InnerRegularWRT.measurableSet_of_isOpen
theorem measurableSet_of_isOpen [OuterRegular μ] (H : InnerRegularWRT μ p IsOpen)
    (hd : ∀ ⦃s U⦄, p s → IsOpen U → p (s \ U)) :
    InnerRegularWRT μ p fun s => MeasurableSet s ∧ μ s ≠ ∞ 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} {p : Set α → Prop} [μ.OuterRegular] (H : μ.InnerRegularWRT p IsOpen) (hd : ∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \ U))
Goal: μ.InnerRegularWRT p fun s => MeasurableSet s ∧ μ s ≠ ⊤
-- MeasureTheory.Measure.InnerRegularWRT.weaklyRegular_of_finite
theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasure μ]
    (H : InnerRegularWRT μ IsClosed IsOpen) : WeaklyRegular μ 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] [BorelSpace α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (H : μ.InnerRegularWRT IsClosed IsOpen)
Goal: μ.WeaklyRegular
-- MeasureTheory.Measure.InnerRegularWRT.of_restrict
lemma of_restrict {μ : Measure α} {s : ℕ → Set α}
    (h : ∀ n, InnerRegularWRT (μ.restrict (s n)) p MeasurableSet)
    (hs : univ ⊆ ⋃ n, s n) (hmono : Monotone s) : InnerRegularWRT μ p MeasurableSet 
Arguments: {α : Type u_1} [MeasurableSpace α] {p : Set α → Prop} {μ : MeasureTheory.Measure α} {s : ℕ → Set α} (h : ∀ (n : ℕ), (μ.restrict (s n)).InnerRegularWRT p MeasurableSet) (hs : Set.univ ⊆ ⋃ n, s n) (hmono : Monotone s)
Goal: μ.InnerRegularWRT p MeasurableSet
-- MeasureTheory.Measure.InnerRegularWRT.of_pseudoMetrizableSpace
theorem of_pseudoMetrizableSpace {X : Type*} [TopologicalSpace X] [PseudoMetrizableSpace X]
    [MeasurableSpace X] (μ : Measure X) : InnerRegularWRT μ IsClosed IsOpen 
Arguments: {X : Type u_3} [TopologicalSpace X] [TopologicalSpace.PseudoMetrizableSpace X] [MeasurableSpace X] (μ : MeasureTheory.Measure X)
Goal: μ.InnerRegularWRT IsClosed IsOpen
-- MeasureTheory.Measure.InnerRegularWRT.isCompact_isClosed
theorem isCompact_isClosed {X : Type*} [TopologicalSpace X] [SigmaCompactSpace X]
    [MeasurableSpace X] (μ : Measure X) : InnerRegularWRT μ IsCompact IsClosed 
Arguments: {X : Type u_3} [TopologicalSpace X] [SigmaCompactSpace X] [MeasurableSpace X] (μ : MeasureTheory.Measure X)
Goal: μ.InnerRegularWRT IsCompact IsClosed
-- MeasureTheory.Measure.InnerRegularWRT.restrict
lemma restrict (h : InnerRegularWRT μ p (fun s ↦ MeasurableSet s ∧ μ s ≠ ∞)) (A : Set α) :
    InnerRegularWRT (μ.restrict A) p (fun s ↦ MeasurableSet s ∧ μ.restrict A s ≠ ∞) 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {p : Set α → Prop} (h : μ.InnerRegularWRT p fun s => MeasurableSet s ∧ μ s ≠ ⊤) (A : Set α)
Goal: (μ.restrict A).InnerRegularWRT p fun s => MeasurableSet s ∧ (μ.restrict A) s ≠ ⊤
-- MeasureTheory.Measure.InnerRegularWRT.restrict_of_measure_ne_top
lemma restrict_of_measure_ne_top (h : InnerRegularWRT μ p (fun s ↦ MeasurableSet s ∧ μ s ≠ ∞))
    {A : Set α} (hA : μ A ≠ ∞) :
    InnerRegularWRT (μ.restrict A) p (fun s ↦ MeasurableSet s) 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {p : Set α → Prop} (h : μ.InnerRegularWRT p fun s => MeasurableSet s ∧ μ s ≠ ⊤) {A : Set α} (hA : μ A ≠ ⊤)
Goal: (μ.restrict A).InnerRegularWRT p fun s => MeasurableSet s
-- MeasureTheory.Measure.InnerRegularWRT.of_sigmaFinite
lemma of_sigmaFinite [SigmaFinite μ] :
    InnerRegularWRT μ (fun s ↦ MeasurableSet s ∧ μ s ≠ ∞) (fun s ↦ MeasurableSet s) 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ]
Goal: μ.InnerRegularWRT (fun s => MeasurableSet s ∧ μ s ≠ ⊤) fun s => MeasurableSet s
-- MeasurableSet.measure_eq_iSup_isCompact
theorem _root_.MeasurableSet.measure_eq_iSup_isCompact ⦃U : Set α⦄ (hU : MeasurableSet U)
    (μ : Measure α) [InnerRegular μ] :
    μ U = ⨆ (K : Set α) (_ : K ⊆ U) (_ : IsCompact K), μ K 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] ⦃U : Set α⦄ (hU : MeasurableSet U) (μ : MeasureTheory.Measure α) [μ.InnerRegular]
Goal: μ U = ⨆ K, ⨆ (_ : K ⊆ U), ⨆ (_ : IsCompact K), μ K
-- MeasureTheory.Measure.InnerRegular.innerRegularWRT_isClosed_isOpen
lemma innerRegularWRT_isClosed_isOpen [R1Space α] [OpensMeasurableSpace α] [h : InnerRegular μ] :
    InnerRegularWRT μ IsClosed IsOpen 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [R1Space α] [OpensMeasurableSpace α] [h : μ.InnerRegular]
Goal: μ.InnerRegularWRT IsClosed IsOpen
-- MeasureTheory.Measure.InnerRegular.exists_compact_not_null
theorem exists_compact_not_null [InnerRegular μ] : (∃ K, IsCompact K ∧ μ K ≠ 0) ↔ μ ≠ 0 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [μ.InnerRegular]
Goal: (∃ K, IsCompact K ∧ μ K ≠ 0) ↔ μ ≠ 0
-- MeasurableSet.exists_lt_isCompact
theorem _root_.MeasurableSet.exists_lt_isCompact [InnerRegular μ] ⦃A : Set α⦄
    (hA : MeasurableSet A) {r : ℝ≥0∞} (hr : r < μ A) :
    ∃ K, K ⊆ A ∧ IsCompact K ∧ r < μ K 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [μ.InnerRegular] ⦃A : Set α⦄ (hA : MeasurableSet A) {r : ℝ≥0∞} (hr : r < μ A)
Goal: ∃ K ⊆ A, IsCompact K ∧ r < μ K
-- MeasureTheory.Measure.InnerRegular.map_of_continuous
protected theorem map_of_continuous [BorelSpace α] [MeasurableSpace β] [TopologicalSpace β]
    [BorelSpace β] [h : InnerRegular μ] {f : α → β} (hf : Continuous f) :
    InnerRegular (Measure.map f μ) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [BorelSpace α] [MeasurableSpace β] [TopologicalSpace β] [BorelSpace β] [h : μ.InnerRegular] {f : α → β} (hf : Continuous f)
Goal: (MeasureTheory.Measure.map f μ).InnerRegular
-- MeasureTheory.Measure.InnerRegular.map
protected theorem map [BorelSpace α] [MeasurableSpace β] [TopologicalSpace β]
    [BorelSpace β] [InnerRegular μ] (f : α ≃ₜ β) : (Measure.map f μ).InnerRegular 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [BorelSpace α] [MeasurableSpace β] [TopologicalSpace β] [BorelSpace β] [μ.InnerRegular] (f : α ≃ₜ β)
Goal: (MeasureTheory.Measure.map (⇑f) μ).InnerRegular
-- MeasureTheory.Measure.InnerRegular.map_iff
protected theorem map_iff [BorelSpace α] [MeasurableSpace β] [TopologicalSpace β]
    [BorelSpace β] (f : α ≃ₜ β) :
    InnerRegular (Measure.map f μ) ↔ InnerRegular μ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [BorelSpace α] [MeasurableSpace β] [TopologicalSpace β] [BorelSpace β] (f : α ≃ₜ β)
Goal: (MeasureTheory.Measure.map (⇑f) μ).InnerRegular ↔ μ.InnerRegular
-- MeasurableSet.exists_isCompact_lt_add
theorem _root_.MeasurableSet.exists_isCompact_lt_add [InnerRegularCompactLTTop μ]
    ⦃A : Set α⦄ (hA : MeasurableSet A) (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∃ K, K ⊆ A ∧ IsCompact K ∧ μ A < μ K + ε 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [μ.InnerRegularCompactLTTop] ⦃A : Set α⦄ (hA : MeasurableSet A) (h'A : μ A ≠ ⊤) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ K ⊆ A, IsCompact K ∧ μ A < μ K + ε
-- MeasurableSet.exists_isCompact_isClosed_lt_add
theorem _root_.MeasurableSet.exists_isCompact_isClosed_lt_add
    [InnerRegularCompactLTTop μ] [R1Space α] [BorelSpace α]
    ⦃A : Set α⦄ (hA : MeasurableSet A) (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∃ K, K ⊆ A ∧ IsCompact K ∧ IsClosed K ∧ μ A < μ K + ε 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [μ.InnerRegularCompactLTTop] [R1Space α] [BorelSpace α] ⦃A : Set α⦄ (hA : MeasurableSet A) (h'A : μ A ≠ ⊤) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ K ⊆ A, IsCompact K ∧ IsClosed K ∧ μ A < μ K + ε
-- MeasurableSet.exists_isCompact_diff_lt
theorem _root_.MeasurableSet.exists_isCompact_diff_lt [OpensMeasurableSpace α] [T2Space α]
    [InnerRegularCompactLTTop μ]  ⦃A : Set α⦄ (hA : MeasurableSet A) (h'A : μ A ≠ ∞)
    {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∃ K, K ⊆ A ∧ IsCompact K ∧ μ (A \ K) < ε 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [OpensMeasurableSpace α] [T2Space α] [μ.InnerRegularCompactLTTop] ⦃A : Set α⦄ (hA : MeasurableSet A) (h'A : μ A ≠ ⊤) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ K ⊆ A, IsCompact K ∧ μ (A \ K) < ε
-- MeasurableSet.exists_isCompact_isClosed_diff_lt
theorem _root_.MeasurableSet.exists_isCompact_isClosed_diff_lt [BorelSpace α] [R1Space α]
    [InnerRegularCompactLTTop μ] ⦃A : Set α⦄ (hA : MeasurableSet A) (h'A : μ A ≠ ∞)
    {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∃ K, K ⊆ A ∧ IsCompact K ∧ IsClosed K ∧ μ (A \ K) < ε 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [BorelSpace α] [R1Space α] [μ.InnerRegularCompactLTTop] ⦃A : Set α⦄ (hA : MeasurableSet A) (h'A : μ A ≠ ⊤) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ K ⊆ A, IsCompact K ∧ IsClosed K ∧ μ (A \ K) < ε
-- MeasurableSet.exists_lt_isCompact_of_ne_top
theorem _root_.MeasurableSet.exists_lt_isCompact_of_ne_top [InnerRegularCompactLTTop μ] ⦃A : Set α⦄
    (hA : MeasurableSet A) (h'A : μ A ≠ ∞) {r : ℝ≥0∞} (hr : r < μ A) :
    ∃ K, K ⊆ A ∧ IsCompact K ∧ r < μ K 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [μ.InnerRegularCompactLTTop] ⦃A : Set α⦄ (hA : MeasurableSet A) (h'A : μ A ≠ ⊤) {r : ℝ≥0∞} (hr : r < μ A)
Goal: ∃ K ⊆ A, IsCompact K ∧ r < μ K
-- MeasurableSet.measure_eq_iSup_isCompact_of_ne_top
theorem _root_.MeasurableSet.measure_eq_iSup_isCompact_of_ne_top [InnerRegularCompactLTTop μ]
    ⦃A : Set α⦄ (hA : MeasurableSet A) (h'A : μ A ≠ ∞) :
    μ A = ⨆ (K) (_ : K ⊆ A) (_ : IsCompact K), μ K 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [μ.InnerRegularCompactLTTop] ⦃A : Set α⦄ (hA : MeasurableSet A) (h'A : μ A ≠ ⊤)
Goal: μ A = ⨆ K, ⨆ (_ : K ⊆ A), ⨆ (_ : IsCompact K), μ K
-- IsCompact.exists_isOpen_lt_of_lt
protected lemma _root_.IsCompact.exists_isOpen_lt_of_lt [InnerRegularCompactLTTop μ]
    [IsLocallyFiniteMeasure μ] [R1Space α] [BorelSpace α] {K : Set α}
    (hK : IsCompact K) (r : ℝ≥0∞) (hr : μ K < r) :
    ∃ U, K ⊆ U ∧ IsOpen U ∧ μ U < r 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [μ.InnerRegularCompactLTTop] [MeasureTheory.IsLocallyFiniteMeasure μ] [R1Space α] [BorelSpace α] {K : Set α} (hK : IsCompact K) (r : ℝ≥0∞) (hr : μ K < r)
Goal: ∃ U, K ⊆ U ∧ IsOpen U ∧ μ U < r
-- IsCompact.measure_eq_iInf_isOpen
protected lemma _root_.IsCompact.measure_eq_iInf_isOpen [InnerRegularCompactLTTop μ]
    [IsLocallyFiniteMeasure μ] [R1Space α] [BorelSpace α] {K : Set α} (hK : IsCompact K) :
    μ K = ⨅ (U : Set α) (_ : K ⊆ U) (_ : IsOpen U), μ U 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [μ.InnerRegularCompactLTTop] [MeasureTheory.IsLocallyFiniteMeasure μ] [R1Space α] [BorelSpace α] {K : Set α} (hK : IsCompact K)
Goal: μ K = ⨅ U, ⨅ (_ : K ⊆ U), ⨅ (_ : IsOpen U), μ U
-- IsCompact.exists_isOpen_lt_add
protected theorem _root_.IsCompact.exists_isOpen_lt_add [InnerRegularCompactLTTop μ]
    [IsLocallyFiniteMeasure μ] [R1Space α] [BorelSpace α]
    {K : Set α} (hK : IsCompact K) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∃ U, K ⊆ U ∧ IsOpen U ∧ μ U < μ K + ε 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [μ.InnerRegularCompactLTTop] [MeasureTheory.IsLocallyFiniteMeasure μ] [R1Space α] [BorelSpace α] {K : Set α} (hK : IsCompact K) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ U, K ⊆ U ∧ IsOpen U ∧ μ U < μ K + ε
-- MeasurableSet.exists_isOpen_symmDiff_lt
protected theorem _root_.MeasurableSet.exists_isOpen_symmDiff_lt [InnerRegularCompactLTTop μ]
    [IsLocallyFiniteMeasure μ] [R1Space α] [BorelSpace α]
    {s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∃ U, IsOpen U ∧ μ U < ∞ ∧ μ (U ∆ s) < ε 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [μ.InnerRegularCompactLTTop] [MeasureTheory.IsLocallyFiniteMeasure μ] [R1Space α] [BorelSpace α] {s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ U, IsOpen U ∧ μ U < ⊤ ∧ μ (U ∆ s) < ε
-- MeasureTheory.Measure.InnerRegularCompactLTTop.map_of_continuous
protected theorem map_of_continuous [BorelSpace α] [MeasurableSpace β] [TopologicalSpace β]
    [BorelSpace β] [h : InnerRegularCompactLTTop μ] {f : α → β} (hf : Continuous f) :
    InnerRegularCompactLTTop (Measure.map f μ) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [BorelSpace α] [MeasurableSpace β] [TopologicalSpace β] [BorelSpace β] [h : μ.InnerRegularCompactLTTop] {f : α → β} (hf : Continuous f)
Goal: (MeasureTheory.Measure.map f μ).InnerRegularCompactLTTop
-- IsOpen.exists_lt_isClosed
theorem _root_.IsOpen.exists_lt_isClosed [WeaklyRegular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
    (hr : r < μ U) : ∃ F, F ⊆ U ∧ IsClosed F ∧ r < μ F 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [μ.WeaklyRegular] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞} (hr : r < μ U)
Goal: ∃ F ⊆ U, IsClosed F ∧ r < μ F
-- IsOpen.measure_eq_iSup_isClosed
theorem _root_.IsOpen.measure_eq_iSup_isClosed ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α)
    [WeaklyRegular μ] : μ U = ⨆ (F) (_ : F ⊆ U) (_ : IsClosed F), μ F 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] ⦃U : Set α⦄ (hU : IsOpen U) (μ : MeasureTheory.Measure α) [μ.WeaklyRegular]
Goal: μ U = ⨆ F, ⨆ (_ : F ⊆ U), ⨆ (_ : IsClosed F), μ F
-- MeasureTheory.Measure.WeaklyRegular.innerRegular_measurable
theorem innerRegular_measurable [WeaklyRegular μ] :
    InnerRegularWRT μ IsClosed fun s => MeasurableSet s ∧ μ s ≠ ∞ 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [μ.WeaklyRegular]
Goal: μ.InnerRegularWRT IsClosed fun s => MeasurableSet s ∧ μ s ≠ ⊤
-- MeasurableSet.exists_isClosed_lt_add
theorem _root_.MeasurableSet.exists_isClosed_lt_add [WeaklyRegular μ] {s : Set α}
    (hs : MeasurableSet s) (hμs : μ s ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∃ K, K ⊆ s ∧ IsClosed K ∧ μ s < μ K + ε 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [μ.WeaklyRegular] {s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ K ⊆ s, IsClosed K ∧ μ s < μ K + ε
-- MeasurableSet.exists_isClosed_diff_lt
theorem _root_.MeasurableSet.exists_isClosed_diff_lt [OpensMeasurableSpace α] [WeaklyRegular μ]
    ⦃A : Set α⦄ (hA : MeasurableSet A) (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∃ F, F ⊆ A ∧ IsClosed F ∧ μ (A \ F) < ε 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [OpensMeasurableSpace α] [μ.WeaklyRegular] ⦃A : Set α⦄ (hA : MeasurableSet A) (h'A : μ A ≠ ⊤) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∃ F ⊆ A, IsClosed F ∧ μ (A \ F) < ε
-- MeasurableSet.exists_lt_isClosed_of_ne_top
theorem _root_.MeasurableSet.exists_lt_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set α⦄
    (hA : MeasurableSet A) (h'A : μ A ≠ ∞) {r : ℝ≥0∞} (hr : r < μ A) :
    ∃ K, K ⊆ A ∧ IsClosed K ∧ r < μ K 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [μ.WeaklyRegular] ⦃A : Set α⦄ (hA : MeasurableSet A) (h'A : μ A ≠ ⊤) {r : ℝ≥0∞} (hr : r < μ A)
Goal: ∃ K ⊆ A, IsClosed K ∧ r < μ K
-- MeasurableSet.measure_eq_iSup_isClosed_of_ne_top
theorem _root_.MeasurableSet.measure_eq_iSup_isClosed_of_ne_top [WeaklyRegular μ] ⦃A : Set α⦄
    (hA : MeasurableSet A) (h'A : μ A ≠ ∞) : μ A = ⨆ (K) (_ : K ⊆ A) (_ : IsClosed K), μ K 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [μ.WeaklyRegular] ⦃A : Set α⦄ (hA : MeasurableSet A) (h'A : μ A ≠ ⊤)
Goal: μ A = ⨆ K, ⨆ (_ : K ⊆ A), ⨆ (_ : IsClosed K), μ K
-- MeasureTheory.Measure.WeaklyRegular.restrict_of_measure_ne_top
theorem restrict_of_measure_ne_top [BorelSpace α] [WeaklyRegular μ] {A : Set α}
    (h'A : μ A ≠ ∞) : WeaklyRegular (μ.restrict A) 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [BorelSpace α] [μ.WeaklyRegular] {A : Set α} (h'A : μ A ≠ ⊤)
Goal: (μ.restrict A).WeaklyRegular
-- MeasureTheory.Measure.WeaklyRegular.smul
protected theorem smul [WeaklyRegular μ] {x : ℝ≥0∞} (hx : x ≠ ∞) : (x • μ).WeaklyRegular 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [μ.WeaklyRegular] {x : ℝ≥0∞} (hx : x ≠ ⊤)
Goal: (x • μ).WeaklyRegular
-- IsOpen.exists_lt_isCompact
theorem _root_.IsOpen.exists_lt_isCompact [Regular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
    (hr : r < μ U) : ∃ K, K ⊆ U ∧ IsCompact K ∧ r < μ K 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [μ.Regular] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞} (hr : r < μ U)
Goal: ∃ K ⊆ U, IsCompact K ∧ r < μ K
-- IsOpen.measure_eq_iSup_isCompact
theorem _root_.IsOpen.measure_eq_iSup_isCompact ⦃U : Set α⦄ (hU : IsOpen U) (μ : Measure α)
    [Regular μ] : μ U = ⨆ (K : Set α) (_ : K ⊆ U) (_ : IsCompact K), μ K 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] ⦃U : Set α⦄ (hU : IsOpen U) (μ : MeasureTheory.Measure α) [μ.Regular]
Goal: μ U = ⨆ K, ⨆ (_ : K ⊆ U), ⨆ (_ : IsCompact K), μ K
-- MeasureTheory.Measure.Regular.exists_compact_not_null
theorem exists_compact_not_null [Regular μ] : (∃ K, IsCompact K ∧ μ K ≠ 0) ↔ μ ≠ 0 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [μ.Regular]
Goal: (∃ K, IsCompact K ∧ μ K ≠ 0) ↔ μ ≠ 0
-- MeasureTheory.Measure.Regular.map
protected theorem map [BorelSpace α] [MeasurableSpace β] [TopologicalSpace β]
    [BorelSpace β] [Regular μ] (f : α ≃ₜ β) : (Measure.map f μ).Regular 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [BorelSpace α] [MeasurableSpace β] [TopologicalSpace β] [BorelSpace β] [μ.Regular] (f : α ≃ₜ β)
Goal: (MeasureTheory.Measure.map (⇑f) μ).Regular
-- MeasureTheory.Measure.Regular.map_iff
protected theorem map_iff [BorelSpace α] [MeasurableSpace β] [TopologicalSpace β]
    [BorelSpace β] (f : α ≃ₜ β) :
    Regular (Measure.map f μ) ↔ Regular μ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [BorelSpace α] [MeasurableSpace β] [TopologicalSpace β] [BorelSpace β] (f : α ≃ₜ β)
Goal: (MeasureTheory.Measure.map (⇑f) μ).Regular ↔ μ.Regular
-- MeasureTheory.Measure.Regular.smul
protected theorem smul [Regular μ] {x : ℝ≥0∞} (hx : x ≠ ∞) : (x • μ).Regular 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [μ.Regular] {x : ℝ≥0∞} (hx : x ≠ ⊤)
Goal: (x • μ).Regular
-- MeasureTheory.Measure.Regular.restrict_of_measure_ne_top
theorem restrict_of_measure_ne_top [R1Space α] [BorelSpace α] [Regular μ]
    {A : Set α} (h'A : μ A ≠ ∞) : Regular (μ.restrict A) 
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [R1Space α] [BorelSpace α] [μ.Regular] {A : Set α} (h'A : μ A ≠ ⊤)
Goal: (μ.restrict A).Regular
-- Subsingleton.aemeasurable
@[nontriviality, measurability]
theorem Subsingleton.aemeasurable [Subsingleton α] : AEMeasurable f μ 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} [Subsingleton α]
Goal: AEMeasurable f μ
-- aemeasurable_of_subsingleton_codomain
@[nontriviality, measurability]
theorem aemeasurable_of_subsingleton_codomain [Subsingleton β] : AEMeasurable f μ 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} [Subsingleton β]
Goal: AEMeasurable f μ