ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Measure.haar.addCHaar_mem_clAddPrehaar
@[to_additive addCHaar_mem_clAddPrehaar]
theorem chaar_mem_clPrehaar (K₀ : PositiveCompacts G) (V : OpenNhdsOf (1 : G)) :
    chaar K₀ ∈ clPrehaar (K₀ : Set G) V 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (K₀ : TopologicalSpace.PositiveCompacts G) (V : TopologicalSpace.OpenNhdsOf 0)
Goal: MeasureTheory.Measure.haar.addCHaar K₀ ∈ MeasureTheory.Measure.haar.clAddPrehaar (↑K₀) V
-- MeasureTheory.Measure.haar.addCHaar_nonneg
@[to_additive addCHaar_nonneg]
theorem chaar_nonneg (K₀ : PositiveCompacts G) (K : Compacts G) : 0 ≤ chaar K₀ K 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (K₀ : TopologicalSpace.PositiveCompacts G) (K : TopologicalSpace.Compacts G)
Goal: 0 ≤ MeasureTheory.Measure.haar.addCHaar K₀ K
-- MeasureTheory.Measure.haar.chaar_nonneg
@[to_additive addCHaar_nonneg]
theorem chaar_nonneg (K₀ : PositiveCompacts G) (K : Compacts G) : 0 ≤ chaar K₀ K 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] (K₀ : TopologicalSpace.PositiveCompacts G) (K : TopologicalSpace.Compacts G)
Goal: 0 ≤ MeasureTheory.Measure.haar.chaar K₀ K
-- MeasureTheory.Measure.haar.chaar_empty
@[to_additive addCHaar_empty]
theorem chaar_empty (K₀ : PositiveCompacts G) : chaar K₀ ⊥ = 0 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] (K₀ : TopologicalSpace.PositiveCompacts G)
Goal: MeasureTheory.Measure.haar.chaar K₀ ⊥ = 0
-- MeasureTheory.Measure.haar.addCHaar_empty
@[to_additive addCHaar_empty]
theorem chaar_empty (K₀ : PositiveCompacts G) : chaar K₀ ⊥ = 0 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (K₀ : TopologicalSpace.PositiveCompacts G)
Goal: MeasureTheory.Measure.haar.addCHaar K₀ ⊥ = 0
-- MeasureTheory.Measure.haar.addCHaar_self
@[to_additive addCHaar_self]
theorem chaar_self (K₀ : PositiveCompacts G) : chaar K₀ K₀.toCompacts = 1 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (K₀ : TopologicalSpace.PositiveCompacts G)
Goal: MeasureTheory.Measure.haar.addCHaar K₀ K₀.toCompacts = 1
-- MeasureTheory.Measure.haar.chaar_self
@[to_additive addCHaar_self]
theorem chaar_self (K₀ : PositiveCompacts G) : chaar K₀ K₀.toCompacts = 1 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] (K₀ : TopologicalSpace.PositiveCompacts G)
Goal: MeasureTheory.Measure.haar.chaar K₀ K₀.toCompacts = 1
-- MeasureTheory.Measure.haar.chaar_mono
@[to_additive addCHaar_mono]
theorem chaar_mono {K₀ : PositiveCompacts G} {K₁ K₂ : Compacts G} (h : (K₁ : Set G) ⊆ K₂) :
    chaar K₀ K₁ ≤ chaar K₀ K₂ 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} {K₁ K₂ : TopologicalSpace.Compacts G} (h : ↑K₁ ⊆ ↑K₂)
Goal: MeasureTheory.Measure.haar.chaar K₀ K₁ ≤ MeasureTheory.Measure.haar.chaar K₀ K₂
-- MeasureTheory.Measure.haar.addCHaar_mono
@[to_additive addCHaar_mono]
theorem chaar_mono {K₀ : PositiveCompacts G} {K₁ K₂ : Compacts G} (h : (K₁ : Set G) ⊆ K₂) :
    chaar K₀ K₁ ≤ chaar K₀ K₂ 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} {K₁ K₂ : TopologicalSpace.Compacts G} (h : ↑K₁ ⊆ ↑K₂)
Goal: MeasureTheory.Measure.haar.addCHaar K₀ K₁ ≤ MeasureTheory.Measure.haar.addCHaar K₀ K₂
-- MeasureTheory.Measure.haar.addCHaar_sup_le
@[to_additive addCHaar_sup_le]
theorem chaar_sup_le {K₀ : PositiveCompacts G} (K₁ K₂ : Compacts G) :
    chaar K₀ (K₁ ⊔ K₂) ≤ chaar K₀ K₁ + chaar K₀ K₂ 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} (K₁ K₂ : TopologicalSpace.Compacts G)
Goal: MeasureTheory.Measure.haar.addCHaar K₀ (K₁ ⊔ K₂) ≤
  MeasureTheory.Measure.haar.addCHaar K₀ K₁ + MeasureTheory.Measure.haar.addCHaar K₀ K₂
-- MeasureTheory.Measure.haar.chaar_sup_le
@[to_additive addCHaar_sup_le]
theorem chaar_sup_le {K₀ : PositiveCompacts G} (K₁ K₂ : Compacts G) :
    chaar K₀ (K₁ ⊔ K₂) ≤ chaar K₀ K₁ + chaar K₀ K₂ 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} (K₁ K₂ : TopologicalSpace.Compacts G)
Goal: MeasureTheory.Measure.haar.chaar K₀ (K₁ ⊔ K₂) ≤
  MeasureTheory.Measure.haar.chaar K₀ K₁ + MeasureTheory.Measure.haar.chaar K₀ K₂
-- MeasureTheory.Measure.haar.chaar_sup_eq
@[to_additive addCHaar_sup_eq]
theorem chaar_sup_eq {K₀ : PositiveCompacts G}
    {K₁ K₂ : Compacts G} (h : Disjoint K₁.1 K₂.1) (h₂ : IsClosed K₂.1) :
    chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂ 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} {K₁ K₂ : TopologicalSpace.Compacts G} (h : Disjoint K₁.carrier K₂.carrier) (h₂ : IsClosed K₂.carrier)
Goal: MeasureTheory.Measure.haar.chaar K₀ (K₁ ⊔ K₂) =
  MeasureTheory.Measure.haar.chaar K₀ K₁ + MeasureTheory.Measure.haar.chaar K₀ K₂
-- MeasureTheory.Measure.haar.addCHaar_sup_eq
@[to_additive addCHaar_sup_eq]
theorem chaar_sup_eq {K₀ : PositiveCompacts G}
    {K₁ K₂ : Compacts G} (h : Disjoint K₁.1 K₂.1) (h₂ : IsClosed K₂.1) :
    chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂ 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} {K₁ K₂ : TopologicalSpace.Compacts G} (h : Disjoint K₁.carrier K₂.carrier) (h₂ : IsClosed K₂.carrier)
Goal: MeasureTheory.Measure.haar.addCHaar K₀ (K₁ ⊔ K₂) =
  MeasureTheory.Measure.haar.addCHaar K₀ K₁ + MeasureTheory.Measure.haar.addCHaar K₀ K₂
-- MeasureTheory.Measure.haar.is_left_invariant_chaar
@[to_additive is_left_invariant_addCHaar]
theorem is_left_invariant_chaar {K₀ : PositiveCompacts G} (g : G) (K : Compacts G) :
    chaar K₀ (K.map _ <| continuous_mul_left g) = chaar K₀ K 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} (g : G) (K : TopologicalSpace.Compacts G)
Goal: MeasureTheory.Measure.haar.chaar K₀ (TopologicalSpace.Compacts.map (fun b => g * b) ⋯ K) =
  MeasureTheory.Measure.haar.chaar K₀ K
-- MeasureTheory.Measure.haar.is_left_invariant_addCHaar
@[to_additive is_left_invariant_addCHaar]
theorem is_left_invariant_chaar {K₀ : PositiveCompacts G} (g : G) (K : Compacts G) :
    chaar K₀ (K.map _ <| continuous_mul_left g) = chaar K₀ K 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} (g : G) (K : TopologicalSpace.Compacts G)
Goal: MeasureTheory.Measure.haar.addCHaar K₀ (TopologicalSpace.Compacts.map (fun b => g + b) ⋯ K) =
  MeasureTheory.Measure.haar.addCHaar K₀ K
-- MeasureTheory.Measure.haar.haarContent_apply
@[to_additive]
theorem haarContent_apply (K₀ : PositiveCompacts G) (K : Compacts G) :
    haarContent K₀ K = show NNReal from ⟨chaar K₀ K, chaar_nonneg _ _⟩ 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] (K₀ : TopologicalSpace.PositiveCompacts G) (K : TopologicalSpace.Compacts G)
Goal: (fun s => ↑((MeasureTheory.Measure.haar.haarContent K₀).toFun s)) K =
  ↑(let_fun this := ⟨MeasureTheory.Measure.haar.chaar K₀ K, ⋯⟩;
    this)
-- MeasureTheory.Measure.haar.addHaarContent_apply
@[to_additive]
theorem haarContent_apply (K₀ : PositiveCompacts G) (K : Compacts G) :
    haarContent K₀ K = show NNReal from ⟨chaar K₀ K, chaar_nonneg _ _⟩ 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (K₀ : TopologicalSpace.PositiveCompacts G) (K : TopologicalSpace.Compacts G)
Goal: (fun s => ↑((MeasureTheory.Measure.haar.addHaarContent K₀).toFun s)) K =
  ↑(let_fun this := ⟨MeasureTheory.Measure.haar.addCHaar K₀ K, ⋯⟩;
    this)
-- MeasureTheory.Measure.haar.addHaarContent_self
@[to_additive "The variant of `addCHaar_self` for `addHaarContent`."]
theorem haarContent_self {K₀ : PositiveCompacts G} : haarContent K₀ K₀.toCompacts = 1 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] {K₀ : TopologicalSpace.PositiveCompacts G}
Goal: (fun s => ↑((MeasureTheory.Measure.haar.addHaarContent K₀).toFun s)) K₀.toCompacts = 1
-- MeasureTheory.Measure.haar.haarContent_self
@[to_additive "The variant of `addCHaar_self` for `addHaarContent`."]
theorem haarContent_self {K₀ : PositiveCompacts G} : haarContent K₀ K₀.toCompacts = 1 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] {K₀ : TopologicalSpace.PositiveCompacts G}
Goal: (fun s => ↑((MeasureTheory.Measure.haar.haarContent K₀).toFun s)) K₀.toCompacts = 1
-- MeasureTheory.Measure.haar.is_left_invariant_haarContent
@[to_additive "The variant of `is_left_invariant_addCHaar` for `addHaarContent`"]
theorem is_left_invariant_haarContent {K₀ : PositiveCompacts G} (g : G) (K : Compacts G) :
    haarContent K₀ (K.map _ <| continuous_mul_left g) = haarContent K₀ K 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} (g : G) (K : TopologicalSpace.Compacts G)
Goal: (fun s => ↑((MeasureTheory.Measure.haar.haarContent K₀).toFun s)) (TopologicalSpace.Compacts.map (fun b => g * b) ⋯ K) =
  (fun s => ↑((MeasureTheory.Measure.haar.haarContent K₀).toFun s)) K
-- MeasureTheory.Measure.haar.is_left_invariant_addHaarContent
@[to_additive "The variant of `is_left_invariant_addCHaar` for `addHaarContent`"]
theorem is_left_invariant_haarContent {K₀ : PositiveCompacts G} (g : G) (K : Compacts G) :
    haarContent K₀ (K.map _ <| continuous_mul_left g) = haarContent K₀ K 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} (g : G) (K : TopologicalSpace.Compacts G)
Goal: (fun s => ↑((MeasureTheory.Measure.haar.addHaarContent K₀).toFun s))
    (TopologicalSpace.Compacts.map (fun b => g + b) ⋯ K) =
  (fun s => ↑((MeasureTheory.Measure.haar.addHaarContent K₀).toFun s)) K
-- MeasureTheory.Measure.haar.addHaarContent_outerMeasure_self_pos
@[to_additive]
theorem haarContent_outerMeasure_self_pos (K₀ : PositiveCompacts G) :
    0 < (haarContent K₀).outerMeasure K₀ 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (K₀ : TopologicalSpace.PositiveCompacts G)
Goal: 0 < (MeasureTheory.Measure.haar.addHaarContent K₀).outerMeasure ↑K₀
-- MeasureTheory.Measure.haar.haarContent_outerMeasure_self_pos
@[to_additive]
theorem haarContent_outerMeasure_self_pos (K₀ : PositiveCompacts G) :
    0 < (haarContent K₀).outerMeasure K₀ 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] (K₀ : TopologicalSpace.PositiveCompacts G)
Goal: 0 < (MeasureTheory.Measure.haar.haarContent K₀).outerMeasure ↑K₀
-- MeasureTheory.Measure.haar.haarContent_outerMeasure_closure_pos
@[to_additive]
theorem haarContent_outerMeasure_closure_pos (K₀ : PositiveCompacts G) :
    0 < (haarContent K₀).outerMeasure (closure K₀) 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] (K₀ : TopologicalSpace.PositiveCompacts G)
Goal: 0 < (MeasureTheory.Measure.haar.haarContent K₀).outerMeasure (closure ↑K₀)
-- MeasureTheory.Measure.haar.addHaarContent_outerMeasure_closure_pos
@[to_additive]
theorem haarContent_outerMeasure_closure_pos (K₀ : PositiveCompacts G) :
    0 < (haarContent K₀).outerMeasure (closure K₀) 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] (K₀ : TopologicalSpace.PositiveCompacts G)
Goal: 0 < (MeasureTheory.Measure.haar.addHaarContent K₀).outerMeasure (closure ↑K₀)
-- MeasureTheory.Measure.addHaarMeasure_apply
@[to_additive]
theorem haarMeasure_apply {K₀ : PositiveCompacts G} {s : Set G} (hs : MeasurableSet s) :
    haarMeasure K₀ s = (haarContent K₀).outerMeasure s / (haarContent K₀).measure K₀ 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] {K₀ : TopologicalSpace.PositiveCompacts G} {s : Set G} (hs : MeasurableSet s)
Goal: (MeasureTheory.Measure.addHaarMeasure K₀) s =
  (MeasureTheory.Measure.haar.addHaarContent K₀).outerMeasure s /
    (MeasureTheory.Measure.haar.addHaarContent K₀).measure ↑K₀
-- MeasureTheory.Measure.haarMeasure_apply
@[to_additive]
theorem haarMeasure_apply {K₀ : PositiveCompacts G} {s : Set G} (hs : MeasurableSet s) :
    haarMeasure K₀ s = (haarContent K₀).outerMeasure s / (haarContent K₀).measure K₀ 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] {K₀ : TopologicalSpace.PositiveCompacts G} {s : Set G} (hs : MeasurableSet s)
Goal: (MeasureTheory.Measure.haarMeasure K₀) s =
  (MeasureTheory.Measure.haar.haarContent K₀).outerMeasure s / (MeasureTheory.Measure.haar.haarContent K₀).measure ↑K₀
-- MeasureTheory.Measure.addHaarMeasure_self
@[to_additive]
theorem haarMeasure_self {K₀ : PositiveCompacts G} : haarMeasure K₀ K₀ = 1 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] {K₀ : TopologicalSpace.PositiveCompacts G}
Goal: (MeasureTheory.Measure.addHaarMeasure K₀) ↑K₀ = 1
-- MeasureTheory.Measure.haarMeasure_self
@[to_additive]
theorem haarMeasure_self {K₀ : PositiveCompacts G} : haarMeasure K₀ K₀ = 1 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] {K₀ : TopologicalSpace.PositiveCompacts G}
Goal: (MeasureTheory.Measure.haarMeasure K₀) ↑K₀ = 1
-- MeasureTheory.Measure.addHaarMeasure_closure_self
@[to_additive]
theorem haarMeasure_closure_self {K₀ : PositiveCompacts G} : haarMeasure K₀ (closure K₀) = 1 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] {K₀ : TopologicalSpace.PositiveCompacts G}
Goal: (MeasureTheory.Measure.addHaarMeasure K₀) (closure ↑K₀) = 1
-- MeasureTheory.Measure.haarMeasure_closure_self
@[to_additive]
theorem haarMeasure_closure_self {K₀ : PositiveCompacts G} : haarMeasure K₀ (closure K₀) = 1 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] {K₀ : TopologicalSpace.PositiveCompacts G}
Goal: (MeasureTheory.Measure.haarMeasure K₀) (closure ↑K₀) = 1
-- MeasureTheory.Measure.sub_mem_nhds_zero_of_addHaar_pos
@[to_additive
"**Steinhaus Theorem** In any locally compact group `G` with an inner regular Haar measure `μ`,
for any measurable set `E` of positive measure, the set `E - E` is a neighbourhood of `0`."]
theorem div_mem_nhds_one_of_haar_pos (μ : Measure G) [IsHaarMeasure μ] [LocallyCompactSpace G]
    [InnerRegular μ] (E : Set G) (hE : MeasurableSet E) (hEpos : 0 < μ E) :
    E / E ∈ 𝓝 (1 : G) 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] (μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [LocallyCompactSpace G] [μ.InnerRegular] (E : Set G) (hE : MeasurableSet E) (hEpos : 0 < μ E)
Goal: E - E ∈ 𝓝 0
-- MeasureTheory.Measure.div_mem_nhds_one_of_haar_pos
@[to_additive
"**Steinhaus Theorem** In any locally compact group `G` with an inner regular Haar measure `μ`,
for any measurable set `E` of positive measure, the set `E - E` is a neighbourhood of `0`."]
theorem div_mem_nhds_one_of_haar_pos (μ : Measure G) [IsHaarMeasure μ] [LocallyCompactSpace G]
    [InnerRegular μ] (E : Set G) (hE : MeasurableSet E) (hEpos : 0 < μ E) :
    E / E ∈ 𝓝 (1 : G) 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] (μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [LocallyCompactSpace G] [μ.InnerRegular] (E : Set G) (hE : MeasurableSet E) (hEpos : 0 < μ E)
Goal: E / E ∈ 𝓝 1
-- MeasureTheory.Measure.haarMeasure_unique
@[to_additive
"**Uniqueness of left-invariant measures**: In a second-countable locally compact additive group,
  any σ-finite left-invariant measure is a scalar multiple of the additive Haar measure.
  This is slightly weaker than assuming that `μ` is a additive Haar measure (in particular we don't
  require `μ ≠ 0`).
  See also `isAddLeftInvariant_eq_smul_of_regular`
  for a statement not assuming second-countability."]
theorem haarMeasure_unique (μ : Measure G) [SigmaFinite μ] [IsMulLeftInvariant μ]
    (K₀ : PositiveCompacts G) : μ = μ K₀ • haarMeasure K₀ 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] [SecondCountableTopology G] (μ : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite μ] [μ.IsMulLeftInvariant] (K₀ : TopologicalSpace.PositiveCompacts G)
Goal: μ = μ ↑K₀ • MeasureTheory.Measure.haarMeasure K₀
-- MeasureTheory.Measure.addHaarMeasure_unique
@[to_additive
"**Uniqueness of left-invariant measures**: In a second-countable locally compact additive group,
  any σ-finite left-invariant measure is a scalar multiple of the additive Haar measure.
  This is slightly weaker than assuming that `μ` is a additive Haar measure (in particular we don't
  require `μ ≠ 0`).
  See also `isAddLeftInvariant_eq_smul_of_regular`
  for a statement not assuming second-countability."]
theorem haarMeasure_unique (μ : Measure G) [SigmaFinite μ] [IsMulLeftInvariant μ]
    (K₀ : PositiveCompacts G) : μ = μ K₀ • haarMeasure K₀ 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] [SecondCountableTopology G] (μ : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite μ] [μ.IsAddLeftInvariant] (K₀ : TopologicalSpace.PositiveCompacts G)
Goal: μ = μ ↑K₀ • MeasureTheory.Measure.addHaarMeasure K₀
-- MeasureTheory.Measure.addHaarMeasure_eq_iff
@[to_additive]
theorem haarMeasure_eq_iff (K₀ : PositiveCompacts G) (μ : Measure G) [SigmaFinite μ]
    [IsMulLeftInvariant μ] :
    haarMeasure K₀ = μ ↔ μ K₀ = 1 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] [SecondCountableTopology G] (K₀ : TopologicalSpace.PositiveCompacts G) (μ : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite μ] [μ.IsAddLeftInvariant]
Goal: MeasureTheory.Measure.addHaarMeasure K₀ = μ ↔ μ ↑K₀ = 1
-- MeasureTheory.Measure.haarMeasure_eq_iff
@[to_additive]
theorem haarMeasure_eq_iff (K₀ : PositiveCompacts G) (μ : Measure G) [SigmaFinite μ]
    [IsMulLeftInvariant μ] :
    haarMeasure K₀ = μ ↔ μ K₀ = 1 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] [SecondCountableTopology G] (K₀ : TopologicalSpace.PositiveCompacts G) (μ : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite μ] [μ.IsMulLeftInvariant]
Goal: MeasureTheory.Measure.haarMeasure K₀ = μ ↔ μ ↑K₀ = 1
-- MeasureTheory.Measure.regular_of_isAddLeftInvariant
@[to_additive
"To show that an invariant σ-finite measure is regular it is sufficient to show that it is finite on
some compact set with non-empty interior."]
theorem regular_of_isMulLeftInvariant {μ : Measure G} [SigmaFinite μ] [IsMulLeftInvariant μ]
    {K : Set G} (hK : IsCompact K) (h2K : (interior K).Nonempty) (hμK : μ K ≠ ∞) : Regular μ 
Arguments: {G : Type u_1} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] [SecondCountableTopology G] {μ : MeasureTheory.Measure G} [MeasureTheory.SigmaFinite μ] [μ.IsAddLeftInvariant] {K : Set G} (hK : IsCompact K) (h2K : (interior K).Nonempty) (hμK : μ K ≠ ⊤)
Goal: μ.Regular
-- MeasureTheory.Measure.regular_of_isMulLeftInvariant
@[to_additive
"To show that an invariant σ-finite measure is regular it is sufficient to show that it is finite on
some compact set with non-empty interior."]
theorem regular_of_isMulLeftInvariant {μ : Measure G} [SigmaFinite μ] [IsMulLeftInvariant μ]
    {K : Set G} (hK : IsCompact K) (h2K : (interior K).Nonempty) (hμK : μ K ≠ ∞) : Regular μ 
Arguments: {G : Type u_1} [Group G] [TopologicalSpace G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] [SecondCountableTopology G] {μ : MeasureTheory.Measure G} [MeasureTheory.SigmaFinite μ] [μ.IsMulLeftInvariant] {K : Set G} (hK : IsCompact K) (h2K : (interior K).Nonempty) (hμK : μ K ≠ ⊤)
Goal: μ.Regular
-- MeasureTheory.Measure.everywherePosSubset_subset
lemma everywherePosSubset_subset (μ : Measure α) (s : Set α) : μ.everywherePosSubset s ⊆ s 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) (s : Set α)
Goal: μ.everywherePosSubset s ⊆ s
-- MeasureTheory.Measure.exists_isOpen_everywherePosSubset_eq_diff
lemma exists_isOpen_everywherePosSubset_eq_diff (μ : Measure α) (s : Set α) :
    ∃ u, IsOpen u ∧ μ.everywherePosSubset s = s \ u 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) (s : Set α)
Goal: ∃ u, IsOpen u ∧ μ.everywherePosSubset s = s \ u
-- MeasurableSet.everywherePosSubset
protected lemma _root_.MeasurableSet.everywherePosSubset [OpensMeasurableSpace α]
    (hs : MeasurableSet s) :
    MeasurableSet (μ.everywherePosSubset s) 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} [OpensMeasurableSpace α] (hs : MeasurableSet s)
Goal: MeasurableSet (μ.everywherePosSubset s)
-- IsClosed.everywherePosSubset
protected lemma _root_.IsClosed.everywherePosSubset (hs : IsClosed s) :
    IsClosed (μ.everywherePosSubset s) 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (hs : IsClosed s)
Goal: IsClosed (μ.everywherePosSubset s)
-- IsCompact.everywherePosSubset
protected lemma _root_.IsCompact.everywherePosSubset (hs : IsCompact s) :
    IsCompact (μ.everywherePosSubset s) 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (hs : IsCompact s)
Goal: IsCompact (μ.everywherePosSubset s)
-- MeasureTheory.Measure.measure_eq_zero_of_subset_diff_everywherePosSubset
lemma measure_eq_zero_of_subset_diff_everywherePosSubset
    (hk : IsCompact k) (h'k : k ⊆ s \ μ.everywherePosSubset s) : μ k = 0 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s k : Set α} (hk : IsCompact k) (h'k : k ⊆ s \ μ.everywherePosSubset s)
Goal: μ k = 0
-- MeasureTheory.Measure.everywherePosSubset_ae_eq
lemma everywherePosSubset_ae_eq [OpensMeasurableSpace α] [InnerRegular μ] (hs : MeasurableSet s) :
    μ.everywherePosSubset s =ᵐ[μ] s 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} [OpensMeasurableSpace α] [μ.InnerRegular] (hs : MeasurableSet s)
Goal: μ.everywherePosSubset s =ᵐ[μ] s
-- MeasureTheory.Measure.everywherePosSubset_ae_eq_of_measure_ne_top
lemma everywherePosSubset_ae_eq_of_measure_ne_top
    [OpensMeasurableSpace α] [InnerRegularCompactLTTop μ] (hs : MeasurableSet s) (h's : μ s ≠ ∞) :
    μ.everywherePosSubset s =ᵐ[μ] s 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} [OpensMeasurableSpace α] [μ.InnerRegularCompactLTTop] (hs : MeasurableSet s) (h's : μ s ≠ ⊤)
Goal: μ.everywherePosSubset s =ᵐ[μ] s
-- MeasureTheory.Measure.isEverywherePos_everywherePosSubset
lemma isEverywherePos_everywherePosSubset
    [OpensMeasurableSpace α] [InnerRegular μ] (hs : MeasurableSet s) :
    μ.IsEverywherePos (μ.everywherePosSubset s) 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} [OpensMeasurableSpace α] [μ.InnerRegular] (hs : MeasurableSet s)
Goal: μ.IsEverywherePos (μ.everywherePosSubset s)
-- MeasureTheory.Measure.isEverywherePos_everywherePosSubset_of_measure_ne_top
lemma isEverywherePos_everywherePosSubset_of_measure_ne_top
    [OpensMeasurableSpace α] [InnerRegularCompactLTTop μ] (hs : MeasurableSet s) (h's : μ s ≠ ∞) :
    μ.IsEverywherePos (μ.everywherePosSubset s) 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} [OpensMeasurableSpace α] [μ.InnerRegularCompactLTTop] (hs : MeasurableSet s) (h's : μ s ≠ ⊤)
Goal: μ.IsEverywherePos (μ.everywherePosSubset s)
-- MeasureTheory.Measure.IsEverywherePos.smul_measure
lemma IsEverywherePos.smul_measure (hs : IsEverywherePos μ s) {c : ℝ≥0∞} (hc : c ≠ 0) :
    IsEverywherePos (c • μ) s 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (hs : μ.IsEverywherePos s) {c : ℝ≥0∞} (hc : c ≠ 0)
Goal: (c • μ).IsEverywherePos s