ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Measure.measure_inv
@[to_additive (attr := simp)]
theorem measure_inv (μ : Measure G) [IsInvInvariant μ] (A : Set G) : μ A⁻¹ = μ A 
Arguments: {G : Type u_2} [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] (μ : MeasureTheory.Measure G) [μ.IsInvInvariant] (A : Set G)
Goal: μ A⁻¹ = μ A
-- MeasureTheory.Measure.measure_preimage_neg
@[to_additive]
theorem measure_preimage_inv (μ : Measure G) [IsInvInvariant μ] (A : Set G) :
    μ (Inv.inv ⁻¹' A) = μ A 
Arguments: {G : Type u_2} [MeasurableSpace G] [InvolutiveNeg G] [MeasurableNeg G] (μ : MeasureTheory.Measure G) [μ.IsNegInvariant] (A : Set G)
Goal: μ (Neg.neg ⁻¹' A) = μ A
-- MeasureTheory.Measure.measure_preimage_inv
@[to_additive]
theorem measure_preimage_inv (μ : Measure G) [IsInvInvariant μ] (A : Set G) :
    μ (Inv.inv ⁻¹' A) = μ A 
Arguments: {G : Type u_2} [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] (μ : MeasureTheory.Measure G) [μ.IsInvInvariant] (A : Set G)
Goal: μ (Inv.inv ⁻¹' A) = μ A
-- MeasureTheory.Measure.measurePreserving_sub_left
@[to_additive]
theorem measurePreserving_div_left (μ : Measure G) [IsInvInvariant μ] [IsMulLeftInvariant μ]
    (g : G) : MeasurePreserving (fun t => g / t) μ μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [SubtractionMonoid G] [MeasurableAdd G] [MeasurableNeg G] (μ : MeasureTheory.Measure G) [μ.IsNegInvariant] [μ.IsAddLeftInvariant] (g : G)
Goal: MeasureTheory.MeasurePreserving (fun t => g - t) μ μ
-- MeasureTheory.Measure.measurePreserving_div_left
@[to_additive]
theorem measurePreserving_div_left (μ : Measure G) [IsInvInvariant μ] [IsMulLeftInvariant μ]
    (g : G) : MeasurePreserving (fun t => g / t) μ μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [DivisionMonoid G] [MeasurableMul G] [MeasurableInv G] (μ : MeasureTheory.Measure G) [μ.IsInvInvariant] [μ.IsMulLeftInvariant] (g : G)
Goal: MeasureTheory.MeasurePreserving (fun t => g / t) μ μ
-- MeasureTheory.Measure.map_sub_left_eq_self
@[to_additive]
theorem map_div_left_eq_self (μ : Measure G) [IsInvInvariant μ] [IsMulLeftInvariant μ] (g : G) :
    map (fun t => g / t) μ = μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [SubtractionMonoid G] [MeasurableAdd G] [MeasurableNeg G] (μ : MeasureTheory.Measure G) [μ.IsNegInvariant] [μ.IsAddLeftInvariant] (g : G)
Goal: MeasureTheory.Measure.map (fun t => g - t) μ = μ
-- MeasureTheory.Measure.map_div_left_eq_self
@[to_additive]
theorem map_div_left_eq_self (μ : Measure G) [IsInvInvariant μ] [IsMulLeftInvariant μ] (g : G) :
    map (fun t => g / t) μ = μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [DivisionMonoid G] [MeasurableMul G] [MeasurableInv G] (μ : MeasureTheory.Measure G) [μ.IsInvInvariant] [μ.IsMulLeftInvariant] (g : G)
Goal: MeasureTheory.Measure.map (fun t => g / t) μ = μ
-- MeasureTheory.Measure.measurePreserving_add_right_neg
@[to_additive]
theorem measurePreserving_mul_right_inv (μ : Measure G) [IsInvInvariant μ] [IsMulLeftInvariant μ]
    (g : G) : MeasurePreserving (fun t => (g * t)⁻¹) μ μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [SubtractionMonoid G] [MeasurableAdd G] [MeasurableNeg G] (μ : MeasureTheory.Measure G) [μ.IsNegInvariant] [μ.IsAddLeftInvariant] (g : G)
Goal: MeasureTheory.MeasurePreserving (fun t => -(g + t)) μ μ
-- MeasureTheory.Measure.measurePreserving_mul_right_inv
@[to_additive]
theorem measurePreserving_mul_right_inv (μ : Measure G) [IsInvInvariant μ] [IsMulLeftInvariant μ]
    (g : G) : MeasurePreserving (fun t => (g * t)⁻¹) μ μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [DivisionMonoid G] [MeasurableMul G] [MeasurableInv G] (μ : MeasureTheory.Measure G) [μ.IsInvInvariant] [μ.IsMulLeftInvariant] (g : G)
Goal: MeasureTheory.MeasurePreserving (fun t => (g * t)⁻¹) μ μ
-- MeasureTheory.Measure.map_mul_right_inv_eq_self
@[to_additive]
theorem map_mul_right_inv_eq_self (μ : Measure G) [IsInvInvariant μ] [IsMulLeftInvariant μ]
    (g : G) : map (fun t => (g * t)⁻¹) μ = μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [DivisionMonoid G] [MeasurableMul G] [MeasurableInv G] (μ : MeasureTheory.Measure G) [μ.IsInvInvariant] [μ.IsMulLeftInvariant] (g : G)
Goal: MeasureTheory.Measure.map (fun t => (g * t)⁻¹) μ = μ
-- MeasureTheory.Measure.map_add_right_neg_eq_self
@[to_additive]
theorem map_mul_right_inv_eq_self (μ : Measure G) [IsInvInvariant μ] [IsMulLeftInvariant μ]
    (g : G) : map (fun t => (g * t)⁻¹) μ = μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [SubtractionMonoid G] [MeasurableAdd G] [MeasurableNeg G] (μ : MeasureTheory.Measure G) [μ.IsNegInvariant] [μ.IsAddLeftInvariant] (g : G)
Goal: MeasureTheory.Measure.map (fun t => -(g + t)) μ = μ
-- MeasureTheory.Measure.map_div_left_ae
@[to_additive]
theorem map_div_left_ae (μ : Measure G) [IsMulLeftInvariant μ] [IsInvInvariant μ] (x : G) :
    Filter.map (fun t => x / t) (ae μ) = ae μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Group G] [MeasurableMul G] [MeasurableInv G] (μ : MeasureTheory.Measure G) [μ.IsMulLeftInvariant] [μ.IsInvInvariant] (x : G)
Goal: Filter.map (fun t => x / t) (MeasureTheory.ae μ) = MeasureTheory.ae μ
-- MeasureTheory.Measure.map_sub_left_ae
@[to_additive]
theorem map_div_left_ae (μ : Measure G) [IsMulLeftInvariant μ] [IsInvInvariant μ] (x : G) :
    Filter.map (fun t => x / t) (ae μ) = ae μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] [MeasurableNeg G] (μ : MeasureTheory.Measure G) [μ.IsAddLeftInvariant] [μ.IsNegInvariant] (x : G)
Goal: Filter.map (fun t => x - t) (MeasureTheory.ae μ) = MeasureTheory.ae μ
-- MeasureTheory.regular_neg_iff
@[to_additive]
theorem regular_inv_iff : μ.inv.Regular ↔ μ.Regular 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G]
Goal: μ.neg.Regular ↔ μ.Regular
-- MeasureTheory.regular_inv_iff
@[to_additive]
theorem regular_inv_iff : μ.inv.Regular ↔ μ.Regular 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G]
Goal: μ.inv.Regular ↔ μ.Regular
-- MeasureTheory.innerRegular_inv_iff
@[to_additive]
theorem innerRegular_inv_iff : μ.inv.InnerRegular ↔ μ.InnerRegular 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G]
Goal: μ.inv.InnerRegular ↔ μ.InnerRegular
-- MeasureTheory.innerRegular_neg_iff
@[to_additive]
theorem innerRegular_inv_iff : μ.inv.InnerRegular ↔ μ.InnerRegular 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G]
Goal: μ.neg.InnerRegular ↔ μ.InnerRegular
-- MeasureTheory.eventually_nhds_zero_measure_vadd_diff_lt
@[to_additive]
lemma eventually_nhds_one_measure_smul_diff_lt [LocallyCompactSpace G]
    [IsFiniteMeasureOnCompacts μ] [InnerRegularCompactLTTop μ] {k : Set G}
    (hk : IsCompact k) (h'k : IsClosed k) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∀ᶠ g in 𝓝 (1 : G), μ (g • k \ k) < ε 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] [LocallyCompactSpace G] [MeasureTheory.IsFiniteMeasureOnCompacts μ] [μ.InnerRegularCompactLTTop] {k : Set G} (hk : IsCompact k) (h'k : IsClosed k) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∀ᶠ (g : G) in 𝓝 0, μ ((g +ᵥ k) \ k) < ε
-- MeasureTheory.eventually_nhds_one_measure_smul_diff_lt
@[to_additive]
lemma eventually_nhds_one_measure_smul_diff_lt [LocallyCompactSpace G]
    [IsFiniteMeasureOnCompacts μ] [InnerRegularCompactLTTop μ] {k : Set G}
    (hk : IsCompact k) (h'k : IsClosed k) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    ∀ᶠ g in 𝓝 (1 : G), μ (g • k \ k) < ε 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] [LocallyCompactSpace G] [MeasureTheory.IsFiniteMeasureOnCompacts μ] [μ.InnerRegularCompactLTTop] {k : Set G} (hk : IsCompact k) (h'k : IsClosed k) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: ∀ᶠ (g : G) in 𝓝 1, μ (g • k \ k) < ε
-- MeasureTheory.tendsto_measure_smul_diff_isCompact_isClosed
@[to_additive]
lemma tendsto_measure_smul_diff_isCompact_isClosed [LocallyCompactSpace G]
    [IsFiniteMeasureOnCompacts μ] [InnerRegularCompactLTTop μ] {k : Set G}
    (hk : IsCompact k) (h'k : IsClosed k) :
    Tendsto (fun g : G ↦ μ (g • k \ k)) (𝓝 1) (𝓝 0) 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] [LocallyCompactSpace G] [MeasureTheory.IsFiniteMeasureOnCompacts μ] [μ.InnerRegularCompactLTTop] {k : Set G} (hk : IsCompact k) (h'k : IsClosed k)
Goal: Filter.Tendsto (fun g => μ (g • k \ k)) (𝓝 1) (𝓝 0)
-- MeasureTheory.tendsto_measure_vadd_diff_isCompact_isClosed
@[to_additive]
lemma tendsto_measure_smul_diff_isCompact_isClosed [LocallyCompactSpace G]
    [IsFiniteMeasureOnCompacts μ] [InnerRegularCompactLTTop μ] {k : Set G}
    (hk : IsCompact k) (h'k : IsClosed k) :
    Tendsto (fun g : G ↦ μ (g • k \ k)) (𝓝 1) (𝓝 0) 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] [LocallyCompactSpace G] [MeasureTheory.IsFiniteMeasureOnCompacts μ] [μ.InnerRegularCompactLTTop] {k : Set G} (hk : IsCompact k) (h'k : IsClosed k)
Goal: Filter.Tendsto (fun g => μ ((g +ᵥ k) \ k)) (𝓝 0) (𝓝 0)
-- MeasureTheory.isOpenPosMeasure_of_mulLeftInvariant_of_compact
@[to_additive
"If a left-invariant measure gives positive mass to a compact set, then it gives positive mass to
any open set."]
theorem isOpenPosMeasure_of_mulLeftInvariant_of_compact (K : Set G) (hK : IsCompact K)
    (h : μ K ≠ 0) : IsOpenPosMeasure μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] [μ.IsMulLeftInvariant] (K : Set G) (hK : IsCompact K) (h : μ K ≠ 0)
Goal: μ.IsOpenPosMeasure
-- MeasureTheory.isOpenPosMeasure_of_addLeftInvariant_of_compact
@[to_additive
"If a left-invariant measure gives positive mass to a compact set, then it gives positive mass to
any open set."]
theorem isOpenPosMeasure_of_mulLeftInvariant_of_compact (K : Set G) (hK : IsCompact K)
    (h : μ K ≠ 0) : IsOpenPosMeasure μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] [μ.IsAddLeftInvariant] (K : Set G) (hK : IsCompact K) (h : μ K ≠ 0)
Goal: μ.IsOpenPosMeasure
-- MeasureTheory.null_iff_of_isMulLeftInvariant
@[to_additive]
theorem null_iff_of_isMulLeftInvariant [Regular μ] {s : Set G} (hs : IsOpen s) :
    μ s = 0 ↔ s = ∅ ∨ μ = 0 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] [μ.IsMulLeftInvariant] [μ.Regular] {s : Set G} (hs : IsOpen s)
Goal: μ s = 0 ↔ s = ∅ ∨ μ = 0
-- MeasureTheory.null_iff_of_isAddLeftInvariant
@[to_additive]
theorem null_iff_of_isMulLeftInvariant [Regular μ] {s : Set G} (hs : IsOpen s) :
    μ s = 0 ↔ s = ∅ ∨ μ = 0 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] [μ.IsAddLeftInvariant] [μ.Regular] {s : Set G} (hs : IsOpen s)
Goal: μ s = 0 ↔ s = ∅ ∨ μ = 0
-- MeasureTheory.measure_ne_zero_iff_nonempty_of_isAddLeftInvariant
@[to_additive]
theorem measure_ne_zero_iff_nonempty_of_isMulLeftInvariant [Regular μ] (hμ : μ ≠ 0) {s : Set G}
    (hs : IsOpen s) : μ s ≠ 0 ↔ s.Nonempty 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] [μ.IsAddLeftInvariant] [μ.Regular] (hμ : μ ≠ 0) {s : Set G} (hs : IsOpen s)
Goal: μ s ≠ 0 ↔ s.Nonempty
-- MeasureTheory.measure_ne_zero_iff_nonempty_of_isMulLeftInvariant
@[to_additive]
theorem measure_ne_zero_iff_nonempty_of_isMulLeftInvariant [Regular μ] (hμ : μ ≠ 0) {s : Set G}
    (hs : IsOpen s) : μ s ≠ 0 ↔ s.Nonempty 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] [μ.IsMulLeftInvariant] [μ.Regular] (hμ : μ ≠ 0) {s : Set G} (hs : IsOpen s)
Goal: μ s ≠ 0 ↔ s.Nonempty
-- MeasureTheory.measure_pos_iff_nonempty_of_isAddLeftInvariant
@[to_additive]
theorem measure_pos_iff_nonempty_of_isMulLeftInvariant [Regular μ] (h3μ : μ ≠ 0) {s : Set G}
    (hs : IsOpen s) : 0 < μ s ↔ s.Nonempty 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] [μ.IsAddLeftInvariant] [μ.Regular] (h3μ : μ ≠ 0) {s : Set G} (hs : IsOpen s)
Goal: 0 < μ s ↔ s.Nonempty
-- MeasureTheory.measure_pos_iff_nonempty_of_isMulLeftInvariant
@[to_additive]
theorem measure_pos_iff_nonempty_of_isMulLeftInvariant [Regular μ] (h3μ : μ ≠ 0) {s : Set G}
    (hs : IsOpen s) : 0 < μ s ↔ s.Nonempty 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] [μ.IsMulLeftInvariant] [μ.Regular] (h3μ : μ ≠ 0) {s : Set G} (hs : IsOpen s)
Goal: 0 < μ s ↔ s.Nonempty
-- MeasureTheory.measure_lt_top_of_isCompact_of_isAddLeftInvariant
@[to_additive
"If a left-invariant measure gives finite mass to a nonempty open set, then it gives finite mass to
any compact set."]
theorem measure_lt_top_of_isCompact_of_isMulLeftInvariant (U : Set G) (hU : IsOpen U)
    (h'U : U.Nonempty) (h : μ U ≠ ∞) {K : Set G} (hK : IsCompact K) : μ K < ∞ 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] [μ.IsAddLeftInvariant] (U : Set G) (hU : IsOpen U) (h'U : U.Nonempty) (h : μ U ≠ ⊤) {K : Set G} (hK : IsCompact K)
Goal: μ K < ⊤
-- MeasureTheory.measure_lt_top_of_isCompact_of_isMulLeftInvariant
@[to_additive
"If a left-invariant measure gives finite mass to a nonempty open set, then it gives finite mass to
any compact set."]
theorem measure_lt_top_of_isCompact_of_isMulLeftInvariant (U : Set G) (hU : IsOpen U)
    (h'U : U.Nonempty) (h : μ U ≠ ∞) {K : Set G} (hK : IsCompact K) : μ K < ∞ 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] [μ.IsMulLeftInvariant] (U : Set G) (hU : IsOpen U) (h'U : U.Nonempty) (h : μ U ≠ ⊤) {K : Set G} (hK : IsCompact K)
Goal: μ K < ⊤
-- MeasureTheory.measure_lt_top_of_isCompact_of_isMulLeftInvariant'
@[to_additive
"If a left-invariant measure gives finite mass to a set with nonempty interior, then it gives finite
mass to any compact set."]
theorem measure_lt_top_of_isCompact_of_isMulLeftInvariant' {U : Set G}
    (hU : (interior U).Nonempty) (h : μ U ≠ ∞) {K : Set G} (hK : IsCompact K) : μ K < ∞ 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] [μ.IsMulLeftInvariant] {U : Set G} (hU : (interior U).Nonempty) (h : μ U ≠ ⊤) {K : Set G} (hK : IsCompact K)
Goal: μ K < ⊤
-- MeasureTheory.measure_lt_top_of_isCompact_of_isAddLeftInvariant'
@[to_additive
"If a left-invariant measure gives finite mass to a set with nonempty interior, then it gives finite
mass to any compact set."]
theorem measure_lt_top_of_isCompact_of_isMulLeftInvariant' {U : Set G}
    (hU : (interior U).Nonempty) (h : μ U ≠ ∞) {K : Set G} (hK : IsCompact K) : μ K < ∞ 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] [μ.IsAddLeftInvariant] {U : Set G} (hU : (interior U).Nonempty) (h : μ U ≠ ⊤) {K : Set G} (hK : IsCompact K)
Goal: μ K < ⊤
-- MeasureTheory.measure_univ_of_isMulLeftInvariant
@[to_additive (attr := simp)
"In a noncompact locally compact additive group, a left-invariant measure which is positive on open
sets has infinite mass."]
theorem measure_univ_of_isMulLeftInvariant [WeaklyLocallyCompactSpace G] [NoncompactSpace G]
    (μ : Measure G) [IsOpenPosMeasure μ] [μ.IsMulLeftInvariant] : μ univ = ∞ 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] [Group G] [TopologicalGroup G] [WeaklyLocallyCompactSpace G] [NoncompactSpace G] (μ : MeasureTheory.Measure G) [μ.IsOpenPosMeasure] [μ.IsMulLeftInvariant]
Goal: μ Set.univ = ⊤
-- MeasureTheory.measure_univ_of_isAddLeftInvariant
@[to_additive (attr := simp)
"In a noncompact locally compact additive group, a left-invariant measure which is positive on open
sets has infinite mass."]
theorem measure_univ_of_isMulLeftInvariant [WeaklyLocallyCompactSpace G] [NoncompactSpace G]
    (μ : Measure G) [IsOpenPosMeasure μ] [μ.IsMulLeftInvariant] : μ univ = ∞ 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] [AddGroup G] [TopologicalAddGroup G] [WeaklyLocallyCompactSpace G] [NoncompactSpace G] (μ : MeasureTheory.Measure G) [μ.IsOpenPosMeasure] [μ.IsAddLeftInvariant]
Goal: μ Set.univ = ⊤
-- MeasurableSet.mul_closure_one_eq
@[to_additive]
lemma _root_.MeasurableSet.mul_closure_one_eq {s : Set G} (hs : MeasurableSet s) :
    s * (closure {1} : Set G) = s 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] [Group G] [TopologicalGroup G] {s : Set G} (hs : MeasurableSet s)
Goal: s * closure {1} = s
-- MeasurableSet.add_closure_zero_eq
@[to_additive]
lemma _root_.MeasurableSet.mul_closure_one_eq {s : Set G} (hs : MeasurableSet s) :
    s * (closure {1} : Set G) = s 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] [AddGroup G] [TopologicalAddGroup G] {s : Set G} (hs : MeasurableSet s)
Goal: s + closure {0} = s
-- IsCompact.closure_subset_of_measurableSet_of_group
@[to_additive (attr := deprecated IsCompact.closure_subset_measurableSet (since := "2024-01-28"))]
lemma _root_.IsCompact.closure_subset_of_measurableSet_of_group {k s : Set G}
    (hk : IsCompact k) (hs : MeasurableSet s) (h : k ⊆ s) : closure k ⊆ s 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] [Group G] [TopologicalGroup G] {k s : Set G} (hk : IsCompact k) (hs : MeasurableSet s) (h : k ⊆ s)
Goal: closure k ⊆ s
-- IsCompact.closure_subset_of_measurableSet_of_addGroup
@[to_additive (attr := deprecated IsCompact.closure_subset_measurableSet (since := "2024-01-28"))]
lemma _root_.IsCompact.closure_subset_of_measurableSet_of_group {k s : Set G}
    (hk : IsCompact k) (hs : MeasurableSet s) (h : k ⊆ s) : closure k ⊆ s 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] [AddGroup G] [TopologicalAddGroup G] {k s : Set G} (hk : IsCompact k) (hs : MeasurableSet s) (h : k ⊆ s)
Goal: closure k ⊆ s
-- MeasureTheory.measure_mul_closure_one
@[to_additive (attr := simp)]
lemma measure_mul_closure_one (s : Set G) (μ : Measure G) :
    μ (s * (closure {1} : Set G)) = μ s 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] [Group G] [TopologicalGroup G] (s : Set G) (μ : MeasureTheory.Measure G)
Goal: μ (s * closure {1}) = μ s
-- MeasureTheory.measure_add_closure_zero
@[to_additive (attr := simp)]
lemma measure_mul_closure_one (s : Set G) (μ : Measure G) :
    μ (s * (closure {1} : Set G)) = μ s 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] [AddGroup G] [TopologicalAddGroup G] (s : Set G) (μ : MeasureTheory.Measure G)
Goal: μ (s + closure {0}) = μ s
-- IsCompact.measure_closure_eq_of_addGroup
@[to_additive (attr := deprecated IsCompact.measure_closure (since := "2024-01-28"))]
lemma _root_.IsCompact.measure_closure_eq_of_group {k : Set G} (hk : IsCompact k) (μ : Measure G) :
    μ (closure k) = μ k 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] [AddGroup G] [TopologicalAddGroup G] {k : Set G} (hk : IsCompact k) (μ : MeasureTheory.Measure G)
Goal: μ (closure k) = μ k
-- IsCompact.measure_closure_eq_of_group
@[to_additive (attr := deprecated IsCompact.measure_closure (since := "2024-01-28"))]
lemma _root_.IsCompact.measure_closure_eq_of_group {k : Set G} (hk : IsCompact k) (μ : Measure G) :
    μ (closure k) = μ k 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] [Group G] [TopologicalGroup G] {k : Set G} (hk : IsCompact k) (μ : MeasureTheory.Measure G)
Goal: μ (closure k) = μ k
-- MeasureTheory.innerRegularWRT_isCompact_isClosed_measure_ne_top_of_addGroup
@[to_additive]
lemma innerRegularWRT_isCompact_isClosed_measure_ne_top_of_group [h : InnerRegularCompactLTTop μ] :
    InnerRegularWRT μ (fun s ↦ IsCompact s ∧ IsClosed s) (fun s ↦ MeasurableSet s ∧ μ s ≠ ∞) 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalAddGroup G] [h : μ.InnerRegularCompactLTTop]
Goal: μ.InnerRegularWRT (fun s => IsCompact s ∧ IsClosed s) fun s => MeasurableSet s ∧ μ s ≠ ⊤
-- MeasureTheory.innerRegularWRT_isCompact_isClosed_measure_ne_top_of_group
@[to_additive]
lemma innerRegularWRT_isCompact_isClosed_measure_ne_top_of_group [h : InnerRegularCompactLTTop μ] :
    InnerRegularWRT μ (fun s ↦ IsCompact s ∧ IsClosed s) (fun s ↦ MeasurableSet s ∧ μ s ≠ ∞) 
Arguments: {G : Type u_2} [MeasurableSpace G] [TopologicalSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [Group G] [TopologicalGroup G] [h : μ.InnerRegularCompactLTTop]
Goal: μ.InnerRegularWRT (fun s => IsCompact s ∧ IsClosed s) fun s => MeasurableSet s ∧ μ s ≠ ⊤
-- MeasureTheory.Measure.haar_singleton
@[to_additive (attr := simp)]
theorem haar_singleton [TopologicalGroup G] [BorelSpace G] (g : G) : μ {g} = μ {(1 : G)} 
Arguments: {G : Type u_2} [MeasurableSpace G] [Group G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [TopologicalGroup G] [BorelSpace G] (g : G)
Goal: μ {g} = μ {1}
-- MeasureTheory.Measure.addHaar_singleton
@[to_additive (attr := simp)]
theorem haar_singleton [TopologicalGroup G] [BorelSpace G] (g : G) : μ {g} = μ {(1 : G)} 
Arguments: {G : Type u_2} [MeasurableSpace G] [AddGroup G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [TopologicalAddGroup G] [BorelSpace G] (g : G)
Goal: μ {g} = μ {0}
-- MeasureTheory.Measure.IsHaarMeasure.smul
@[to_additive IsAddHaarMeasure.smul]
theorem IsHaarMeasure.smul {c : ℝ≥0∞} (cpos : c ≠ 0) (ctop : c ≠ ∞) : IsHaarMeasure (c • μ) 
Arguments: {G : Type u_2} [MeasurableSpace G] [Group G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] {c : ℝ≥0∞} (cpos : c ≠ 0) (ctop : c ≠ ⊤)
Goal: (c • μ).IsHaarMeasure
-- MeasureTheory.Measure.IsAddHaarMeasure.smul
@[to_additive IsAddHaarMeasure.smul]
theorem IsHaarMeasure.smul {c : ℝ≥0∞} (cpos : c ≠ 0) (ctop : c ≠ ∞) : IsHaarMeasure (c • μ) 
Arguments: {G : Type u_2} [MeasurableSpace G] [AddGroup G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] {c : ℝ≥0∞} (cpos : c ≠ 0) (ctop : c ≠ ⊤)
Goal: (c • μ).IsAddHaarMeasure
-- MeasureTheory.Measure.isHaarMeasure_of_isCompact_nonempty_interior
@[to_additive
"If a left-invariant measure gives positive mass to some compact set with nonempty interior, then
it is an additive Haar measure."]
theorem isHaarMeasure_of_isCompact_nonempty_interior [TopologicalGroup G] [BorelSpace G]
    (μ : Measure G) [IsMulLeftInvariant μ] (K : Set G) (hK : IsCompact K)
    (h'K : (interior K).Nonempty) (h : μ K ≠ 0) (h' : μ K ≠ ∞) : IsHaarMeasure μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [Group G] [TopologicalSpace G] [TopologicalGroup G] [BorelSpace G] (μ : MeasureTheory.Measure G) [μ.IsMulLeftInvariant] (K : Set G) (hK : IsCompact K) (h'K : (interior K).Nonempty) (h : μ K ≠ 0) (h' : μ K ≠ ⊤)
Goal: μ.IsHaarMeasure