ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.toFiniteAux_eq_zero_iff
lemma toFiniteAux_eq_zero_iff [SFinite μ] : μ.toFiniteAux = 0 ↔ μ = 0 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SFinite μ]
Goal: μ.toFiniteAux = 0 ↔ μ = 0
-- MeasureTheory.toFiniteAux_univ_le_one
lemma toFiniteAux_univ_le_one (μ : Measure α) [SFinite μ] : μ.toFiniteAux Set.univ ≤ 1 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ]
Goal: μ.toFiniteAux Set.univ ≤ 1
-- MeasureTheory.toFinite_eq_zero_iff
@[simp]
lemma toFinite_eq_zero_iff [SFinite μ] : μ.toFinite = 0 ↔ μ = 0 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SFinite μ]
Goal: μ.toFinite = 0 ↔ μ = 0
-- MeasureTheory.sFiniteSeq_absolutelyContinuous_toFiniteAux
lemma sFiniteSeq_absolutelyContinuous_toFiniteAux (μ : Measure α) [SFinite μ] (n : ℕ) :
    sFiniteSeq μ n ≪ μ.toFiniteAux 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] (n : ℕ)
Goal: (MeasureTheory.sFiniteSeq μ n).AbsolutelyContinuous μ.toFiniteAux
-- MeasureTheory.toFiniteAux_absolutelyContinuous_toFinite
lemma toFiniteAux_absolutelyContinuous_toFinite (μ : Measure α) [SFinite μ] :
    μ.toFiniteAux ≪ μ.toFinite 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ]
Goal: μ.toFiniteAux.AbsolutelyContinuous μ.toFinite
-- MeasureTheory.sFiniteSeq_absolutelyContinuous_toFinite
lemma sFiniteSeq_absolutelyContinuous_toFinite (μ : Measure α) [SFinite μ] (n : ℕ) :
    sFiniteSeq μ n ≪ μ.toFinite 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] (n : ℕ)
Goal: (MeasureTheory.sFiniteSeq μ n).AbsolutelyContinuous μ.toFinite
-- MeasureTheory.absolutelyContinuous_toFinite
lemma absolutelyContinuous_toFinite (μ : Measure α) [SFinite μ] : μ ≪ μ.toFinite 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ]
Goal: μ.AbsolutelyContinuous μ.toFinite
-- MeasureTheory.toFinite_absolutelyContinuous
lemma toFinite_absolutelyContinuous (μ : Measure α) [SFinite μ] : μ.toFinite ≪ μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ]
Goal: μ.toFinite.AbsolutelyContinuous μ
-- MeasureTheory.densityToFinite_def
lemma densityToFinite_def (μ : Measure α) [SFinite μ] :
    μ.densityToFinite = fun a ↦ ∑' n, (sFiniteSeq μ n).rnDeriv μ.toFinite a 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ]
Goal: μ.densityToFinite = fun a => ∑' (n : ℕ), (MeasureTheory.sFiniteSeq μ n).rnDeriv μ.toFinite a
-- MeasureTheory.measurable_densityToFinite
lemma measurable_densityToFinite (μ : Measure α) [SFinite μ] : Measurable μ.densityToFinite 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ]
Goal: Measurable μ.densityToFinite
-- MeasureTheory.withDensity_densitytoFinite
theorem withDensity_densitytoFinite (μ : Measure α) [SFinite μ] :
    μ.toFinite.withDensity μ.densityToFinite = μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ]
Goal: μ.toFinite.withDensity μ.densityToFinite = μ
-- MeasureTheory.densityToFinite_ae_lt_top
lemma densityToFinite_ae_lt_top (μ : Measure α) [SigmaFinite μ] :
    ∀ᵐ x ∂μ, μ.densityToFinite x < ∞ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ]
Goal: ∀ᵐ (x : α) ∂μ, μ.densityToFinite x < ⊤
-- MeasureTheory.densityToFinite_ae_ne_top
lemma densityToFinite_ae_ne_top (μ : Measure α) [SigmaFinite μ] :
    ∀ᵐ x ∂μ, μ.densityToFinite x ≠ ∞ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ]
Goal: ∀ᵐ (x : α) ∂μ, μ.densityToFinite x ≠ ⊤
-- MeasureTheory.measurableSet_sigmaFiniteSet
lemma measurableSet_sigmaFiniteSet (μ : Measure α) [SFinite μ] :
    MeasurableSet μ.sigmaFiniteSet 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ]
Goal: MeasurableSet μ.sigmaFiniteSet
-- MeasureTheory.restrict_compl_sigmaFiniteSet
lemma restrict_compl_sigmaFiniteSet [SFinite μ] :
    μ.restrict μ.sigmaFiniteSetᶜ = ∞ • μ.toFinite.restrict μ.sigmaFiniteSetᶜ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SFinite μ]
Goal: μ.restrict μ.sigmaFiniteSetᶜ = ⊤ • μ.toFinite.restrict μ.sigmaFiniteSetᶜ
-- MeasureTheory.restrict_compl_sigmaFiniteSet_eq_zero_or_top
lemma restrict_compl_sigmaFiniteSet_eq_zero_or_top (μ : Measure α) [SFinite μ] (s : Set α) :
    μ.restrict μ.sigmaFiniteSetᶜ s = 0 ∨ μ.restrict μ.sigmaFiniteSetᶜ s = ∞ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] (s : Set α)
Goal: (μ.restrict μ.sigmaFiniteSetᶜ) s = 0 ∨ (μ.restrict μ.sigmaFiniteSetᶜ) s = ⊤
-- MeasureTheory.measure_eq_zero_or_top_of_subset_compl_sigmaFiniteSet
lemma measure_eq_zero_or_top_of_subset_compl_sigmaFiniteSet [SFinite μ]
    (ht_subset : t ⊆ μ.sigmaFiniteSetᶜ) :
    μ t = 0 ∨ μ t = ∞ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set α} [MeasureTheory.SFinite μ] (ht_subset : t ⊆ μ.sigmaFiniteSetᶜ)
Goal: μ t = 0 ∨ μ t = ⊤
-- MeasureTheory.toFinite_withDensity_restrict_sigmaFiniteSet
lemma toFinite_withDensity_restrict_sigmaFiniteSet (μ : Measure α) [SFinite μ] :
    (μ.toFinite.withDensity
        (fun x ↦ if μ.densityToFinite x ≠ ∞ then μ.densityToFinite x else 1)).restrict
        μ.sigmaFiniteSet
      = μ.restrict μ.sigmaFiniteSet 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ]
Goal: (μ.toFinite.withDensity fun x => if μ.densityToFinite x ≠ ⊤ then μ.densityToFinite x else 1).restrict μ.sigmaFiniteSet =
  μ.restrict μ.sigmaFiniteSet
-- MeasureTheory.sigmaFinite_of_measure_compl_sigmaFiniteSet_eq_zero
lemma sigmaFinite_of_measure_compl_sigmaFiniteSet_eq_zero [SFinite μ]
    (h : μ μ.sigmaFiniteSetᶜ = 0) :
    SigmaFinite μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SFinite μ] (h : μ μ.sigmaFiniteSetᶜ = 0)
Goal: MeasureTheory.SigmaFinite μ
-- MeasureTheory.measure_compl_sigmaFiniteSet
@[simp]
lemma measure_compl_sigmaFiniteSet (μ : Measure α) [SigmaFinite μ] : μ μ.sigmaFiniteSetᶜ = 0 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ]
Goal: μ μ.sigmaFiniteSetᶜ = 0
-- MeasureTheory.measure_compl_sigmaFiniteSet_eq_zero_iff_sigmaFinite
lemma measure_compl_sigmaFiniteSet_eq_zero_iff_sigmaFinite (μ : Measure α) [SFinite μ] :
    μ μ.sigmaFiniteSetᶜ = 0 ↔ SigmaFinite μ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ]
Goal: μ μ.sigmaFiniteSetᶜ = 0 ↔ MeasureTheory.SigmaFinite μ
-- MeasureTheory.VAddInvariantMeasure.measure_preimage_vadd
class VAddInvariantMeasure (M α : Type*) [VAdd M α] {_ : MeasurableSpace α} (μ : Measure α) :
  Prop where
  measure_preimage_vadd : ∀ (c : M) ⦃s : Set α⦄, MeasurableSet s → μ ((fun x => c +ᵥ x) ⁻¹' s) = μ s
Arguments: {M : Type u_1} {α : Type u_2} [VAdd M α]
Goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [self : MeasureTheory.VAddInvariantMeasure M α μ] (c : M)
  ⦃s : Set α⦄, MeasurableSet s → μ ((fun x => c +ᵥ x) ⁻¹' s) = μ s
-- MeasureTheory.SMulInvariantMeasure.measure_preimage_smul
@[to_additive]
class SMulInvariantMeasure (M α : Type*) [SMul M α] {_ : MeasurableSpace α} (μ : Measure α) :
  Prop where
  measure_preimage_smul : ∀ (c : M) ⦃s : Set α⦄, MeasurableSet s → μ ((fun x => c • x) ⁻¹' s) = μ s
Arguments: {M : Type u_1} {α : Type u_2} [SMul M α]
Goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [self : MeasureTheory.SMulInvariantMeasure M α μ] (c : M)
  ⦃s : Set α⦄, MeasurableSet s → μ ((fun x => c • x) ⁻¹' s) = μ s
-- MeasureTheory.measure_preimage_smul_le
@[to_additive "See also `measure_preimage_smul_of_nullMeasurableSet` and `measure_preimage_smul`."]
theorem measure_preimage_smul_le (c : G) (s : Set α) : μ ((c • ·) ⁻¹' s) ≤ μ s 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [SMul G α] (μ : MeasureTheory.Measure α) [MeasureTheory.SMulInvariantMeasure G α μ] (c : G) (s : Set α)
Goal: μ ((fun x => c • x) ⁻¹' s) ≤ μ s
-- MeasureTheory.measure_preimage_vadd_le
@[to_additive "See also `measure_preimage_smul_of_nullMeasurableSet` and `measure_preimage_smul`."]
theorem measure_preimage_smul_le (c : G) (s : Set α) : μ ((c • ·) ⁻¹' s) ≤ μ s 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [VAdd G α] (μ : MeasureTheory.Measure α) [MeasureTheory.VAddInvariantMeasure G α μ] (c : G) (s : Set α)
Goal: μ ((fun x => c +ᵥ x) ⁻¹' s) ≤ μ s
-- MeasureTheory.tendsto_vadd_ae
@[to_additive "See also `vadd_ae`."]
theorem tendsto_smul_ae (c : G) : Filter.Tendsto (c • ·) (ae μ) (ae μ) 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [VAdd G α] (μ : MeasureTheory.Measure α) [MeasureTheory.VAddInvariantMeasure G α μ] (c : G)
Goal: Filter.Tendsto (fun x => c +ᵥ x) (MeasureTheory.ae μ) (MeasureTheory.ae μ)
-- MeasureTheory.tendsto_smul_ae
@[to_additive "See also `vadd_ae`."]
theorem tendsto_smul_ae (c : G) : Filter.Tendsto (c • ·) (ae μ) (ae μ) 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [SMul G α] (μ : MeasureTheory.Measure α) [MeasureTheory.SMulInvariantMeasure G α μ] (c : G)
Goal: Filter.Tendsto (fun x => c • x) (MeasureTheory.ae μ) (MeasureTheory.ae μ)
-- MeasureTheory.measure_preimage_vadd_null
@[to_additive]
theorem measure_preimage_smul_null (h : μ s = 0) (c : G) : μ ((c • ·) ⁻¹' s) = 0 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [VAdd G α] {μ : MeasureTheory.Measure α} [MeasureTheory.VAddInvariantMeasure G α μ] {s : Set α} (h : μ s = 0) (c : G)
Goal: μ ((fun x => c +ᵥ x) ⁻¹' s) = 0
-- MeasureTheory.measure_preimage_smul_null
@[to_additive]
theorem measure_preimage_smul_null (h : μ s = 0) (c : G) : μ ((c • ·) ⁻¹' s) = 0 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [SMul G α] {μ : MeasureTheory.Measure α} [MeasureTheory.SMulInvariantMeasure G α μ] {s : Set α} (h : μ s = 0) (c : G)
Goal: μ ((fun x => c • x) ⁻¹' s) = 0
-- MeasureTheory.measure_preimage_vadd_of_nullMeasurableSet
@[to_additive]
theorem measure_preimage_smul_of_nullMeasurableSet (hs : NullMeasurableSet s μ) (c : G) :
    μ ((c • ·) ⁻¹' s) = μ s 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [VAdd G α] {μ : MeasureTheory.Measure α} [MeasureTheory.VAddInvariantMeasure G α μ] {s : Set α} (hs : MeasureTheory.NullMeasurableSet s μ) (c : G)
Goal: μ ((fun x => c +ᵥ x) ⁻¹' s) = μ s
-- MeasureTheory.measure_preimage_smul_of_nullMeasurableSet
@[to_additive]
theorem measure_preimage_smul_of_nullMeasurableSet (hs : NullMeasurableSet s μ) (c : G) :
    μ ((c • ·) ⁻¹' s) = μ s 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [SMul G α] {μ : MeasureTheory.Measure α} [MeasureTheory.SMulInvariantMeasure G α μ] {s : Set α} (hs : MeasureTheory.NullMeasurableSet s μ) (c : G)
Goal: μ ((fun x => c • x) ⁻¹' s) = μ s
-- MeasureTheory.measure_preimage_vadd
@[to_additive (attr := simp)]
theorem measure_preimage_smul (c : G) (s : Set α) : μ ((c • ·) ⁻¹' s) = μ s 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [AddGroup G] [AddAction G α] (μ : MeasureTheory.Measure α) [MeasureTheory.VAddInvariantMeasure G α μ] (c : G) (s : Set α)
Goal: μ ((fun x => c +ᵥ x) ⁻¹' s) = μ s
-- MeasureTheory.measure_preimage_smul
@[to_additive (attr := simp)]
theorem measure_preimage_smul (c : G) (s : Set α) : μ ((c • ·) ⁻¹' s) = μ s 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [Group G] [MulAction G α] (μ : MeasureTheory.Measure α) [MeasureTheory.SMulInvariantMeasure G α μ] (c : G) (s : Set α)
Goal: μ ((fun x => c • x) ⁻¹' s) = μ s
-- MeasureTheory.measure_smul
@[to_additive (attr := simp)]
theorem measure_smul (c : G) (s : Set α) : μ (c • s) = μ s 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [Group G] [MulAction G α] (μ : MeasureTheory.Measure α) [MeasureTheory.SMulInvariantMeasure G α μ] (c : G) (s : Set α)
Goal: μ (c • s) = μ s
-- MeasureTheory.measure_vadd
@[to_additive (attr := simp)]
theorem measure_smul (c : G) (s : Set α) : μ (c • s) = μ s 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [AddGroup G] [AddAction G α] (μ : MeasureTheory.Measure α) [MeasureTheory.VAddInvariantMeasure G α μ] (c : G) (s : Set α)
Goal: μ (c +ᵥ s) = μ s
-- MeasureTheory.measure_smul_eq_zero_iff
@[to_additive]
theorem measure_smul_eq_zero_iff {s} (c : G) : μ (c • s) = 0 ↔ μ s = 0 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [Group G] [MulAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.SMulInvariantMeasure G α μ] {s : Set α} (c : G)
Goal: μ (c • s) = 0 ↔ μ s = 0
-- MeasureTheory.measure_vadd_eq_zero_iff
@[to_additive]
theorem measure_smul_eq_zero_iff {s} (c : G) : μ (c • s) = 0 ↔ μ s = 0 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [AddGroup G] [AddAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.VAddInvariantMeasure G α μ] {s : Set α} (c : G)
Goal: μ (c +ᵥ s) = 0 ↔ μ s = 0
-- MeasureTheory.measure_vadd_null
@[to_additive]
theorem measure_smul_null {s} (h : μ s = 0) (c : G) : μ (c • s) = 0 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [AddGroup G] [AddAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.VAddInvariantMeasure G α μ] {s : Set α} (h : μ s = 0) (c : G)
Goal: μ (c +ᵥ s) = 0
-- MeasureTheory.measure_smul_null
@[to_additive]
theorem measure_smul_null {s} (h : μ s = 0) (c : G) : μ (c • s) = 0 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [Group G] [MulAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.SMulInvariantMeasure G α μ] {s : Set α} (h : μ s = 0) (c : G)
Goal: μ (c • s) = 0
-- MeasureTheory.vadd_mem_ae
@[to_additive (attr := simp)]
theorem smul_mem_ae (c : G) {s : Set α} : c • s ∈ ae μ ↔ s ∈ ae μ 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [AddGroup G] [AddAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.VAddInvariantMeasure G α μ] (c : G) {s : Set α}
Goal: c +ᵥ s ∈ MeasureTheory.ae μ ↔ s ∈ MeasureTheory.ae μ
-- MeasureTheory.smul_mem_ae
@[to_additive (attr := simp)]
theorem smul_mem_ae (c : G) {s : Set α} : c • s ∈ ae μ ↔ s ∈ ae μ 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [Group G] [MulAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.SMulInvariantMeasure G α μ] (c : G) {s : Set α}
Goal: c • s ∈ MeasureTheory.ae μ ↔ s ∈ MeasureTheory.ae μ
-- MeasureTheory.vadd_ae
@[to_additive (attr := simp)]
theorem smul_ae (c : G) : c • ae μ = ae μ 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [AddGroup G] [AddAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.VAddInvariantMeasure G α μ] (c : G)
Goal: c +ᵥ MeasureTheory.ae μ = MeasureTheory.ae μ
-- MeasureTheory.smul_ae
@[to_additive (attr := simp)]
theorem smul_ae (c : G) : c • ae μ = ae μ 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [Group G] [MulAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.SMulInvariantMeasure G α μ] (c : G)
Goal: c • MeasureTheory.ae μ = MeasureTheory.ae μ
-- MeasureTheory.eventuallyConst_vadd_set_ae
@[to_additive (attr := simp)]
theorem eventuallyConst_smul_set_ae (c : G) {s : Set α} :
    EventuallyConst (c • s : Set α) (ae μ) ↔ EventuallyConst s (ae μ) 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [AddGroup G] [AddAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.VAddInvariantMeasure G α μ] (c : G) {s : Set α}
Goal: Filter.EventuallyConst (c +ᵥ s) (MeasureTheory.ae μ) ↔ Filter.EventuallyConst s (MeasureTheory.ae μ)
-- MeasureTheory.eventuallyConst_smul_set_ae
@[to_additive (attr := simp)]
theorem eventuallyConst_smul_set_ae (c : G) {s : Set α} :
    EventuallyConst (c • s : Set α) (ae μ) ↔ EventuallyConst s (ae μ) 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [Group G] [MulAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.SMulInvariantMeasure G α μ] (c : G) {s : Set α}
Goal: Filter.EventuallyConst (c • s) (MeasureTheory.ae μ) ↔ Filter.EventuallyConst s (MeasureTheory.ae μ)
-- MeasureTheory.vadd_set_ae_le
@[to_additive (attr := simp)]
theorem smul_set_ae_le (c : G) {s t : Set α} : c • s ≤ᵐ[μ] c • t ↔ s ≤ᵐ[μ] t 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [AddGroup G] [AddAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.VAddInvariantMeasure G α μ] (c : G) {s t : Set α}
Goal: c +ᵥ s ≤ᵐ[μ] c +ᵥ t ↔ s ≤ᵐ[μ] t
-- MeasureTheory.smul_set_ae_le
@[to_additive (attr := simp)]
theorem smul_set_ae_le (c : G) {s t : Set α} : c • s ≤ᵐ[μ] c • t ↔ s ≤ᵐ[μ] t 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [Group G] [MulAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.SMulInvariantMeasure G α μ] (c : G) {s t : Set α}
Goal: c • s ≤ᵐ[μ] c • t ↔ s ≤ᵐ[μ] t
-- MeasureTheory.smul_set_ae_eq
@[to_additive (attr := simp)]
theorem smul_set_ae_eq (c : G) {s t : Set α} : c • s =ᵐ[μ] c • t ↔ s =ᵐ[μ] t 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [Group G] [MulAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.SMulInvariantMeasure G α μ] (c : G) {s t : Set α}
Goal: c • s =ᵐ[μ] c • t ↔ s =ᵐ[μ] t
-- MeasureTheory.vadd_set_ae_eq
@[to_additive (attr := simp)]
theorem smul_set_ae_eq (c : G) {s t : Set α} : c • s =ᵐ[μ] c • t ↔ s =ᵐ[μ] t 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [AddGroup G] [AddAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.VAddInvariantMeasure G α μ] (c : G) {s t : Set α}
Goal: c +ᵥ s =ᵐ[μ] c +ᵥ t ↔ s =ᵐ[μ] t
-- MeasureTheory.measurePreserving_vadd
@[to_additive (attr := simp)]
theorem measurePreserving_smul : MeasurePreserving (c • ·) μ μ 
Arguments: {M : Type v} {α : Type w} {m : MeasurableSpace α} [MeasurableSpace M] [VAdd M α] [MeasurableVAdd M α] (c : M) (μ : MeasureTheory.Measure α) [MeasureTheory.VAddInvariantMeasure M α μ]
Goal: MeasureTheory.MeasurePreserving (fun x => c +ᵥ x) μ μ
-- MeasureTheory.measurePreserving_smul
@[to_additive (attr := simp)]
theorem measurePreserving_smul : MeasurePreserving (c • ·) μ μ 
Arguments: {M : Type v} {α : Type w} {m : MeasurableSpace α} [MeasurableSpace M] [SMul M α] [MeasurableSMul M α] (c : M) (μ : MeasureTheory.Measure α) [MeasureTheory.SMulInvariantMeasure M α μ]
Goal: MeasureTheory.MeasurePreserving (fun x => c • x) μ μ
-- MeasureTheory.map_vadd
@[to_additive (attr := simp)]
theorem map_smul : map (c • ·) μ = μ 
Arguments: {M : Type v} {α : Type w} {m : MeasurableSpace α} [MeasurableSpace M] [VAdd M α] [MeasurableVAdd M α] (c : M) (μ : MeasureTheory.Measure α) [MeasureTheory.VAddInvariantMeasure M α μ]
Goal: MeasureTheory.Measure.map (fun x => c +ᵥ x) μ = μ
-- MeasureTheory.map_smul
@[to_additive (attr := simp)]
theorem map_smul : map (c • ·) μ = μ 
Arguments: {M : Type v} {α : Type w} {m : MeasurableSpace α} [MeasurableSpace M] [SMul M α] [MeasurableSMul M α] (c : M) (μ : MeasureTheory.Measure α) [MeasureTheory.SMulInvariantMeasure M α μ]
Goal: MeasureTheory.Measure.map (fun x => c • x) μ = μ
-- MeasureTheory.smulInvariantMeasure_map
@[to_additive]
theorem smulInvariantMeasure_map [SMul M α] [SMul M β]
    [MeasurableSMul M β]
    (μ : Measure α) [SMulInvariantMeasure M α μ] (f : α → β)
    (hsmul : ∀ (m : M) a, f (m • a) = m • f a) (hf : Measurable f) :
    SMulInvariantMeasure M β (map f μ) 
Arguments: {M : Type uM} {α : Type uα} {β : Type uβ} [MeasurableSpace M] [MeasurableSpace α] [MeasurableSpace β] [SMul M α] [SMul M β] [MeasurableSMul M β] (μ : MeasureTheory.Measure α) [MeasureTheory.SMulInvariantMeasure M α μ] (f : α → β) (hsmul : ∀ (m : M) (a : α), f (m • a) = m • f a) (hf : Measurable f)
Goal: MeasureTheory.SMulInvariantMeasure M β (MeasureTheory.Measure.map f μ)
-- MeasureTheory.vaddInvariantMeasure_map
@[to_additive]
theorem smulInvariantMeasure_map [SMul M α] [SMul M β]
    [MeasurableSMul M β]
    (μ : Measure α) [SMulInvariantMeasure M α μ] (f : α → β)
    (hsmul : ∀ (m : M) a, f (m • a) = m • f a) (hf : Measurable f) :
    SMulInvariantMeasure M β (map f μ) 
Arguments: {M : Type uM} {α : Type uα} {β : Type uβ} [MeasurableSpace M] [MeasurableSpace α] [MeasurableSpace β] [VAdd M α] [VAdd M β] [MeasurableVAdd M β] (μ : MeasureTheory.Measure α) [MeasureTheory.VAddInvariantMeasure M α μ] (f : α → β) (hsmul : ∀ (m : M) (a : α), f (m +ᵥ a) = m +ᵥ f a) (hf : Measurable f)
Goal: MeasureTheory.VAddInvariantMeasure M β (MeasureTheory.Measure.map f μ)
-- MeasureTheory.vaddInvariantMeasure_tfae
@[to_additive]
theorem smulInvariantMeasure_tfae :
    List.TFAE
      [SMulInvariantMeasure G α μ,
        ∀ (c : G) (s), MeasurableSet s → μ ((c • ·) ⁻¹' s) = μ s,
        ∀ (c : G) (s), MeasurableSet s → μ (c • s) = μ s,
        ∀ (c : G) (s), μ ((c • ·) ⁻¹' s) = μ s,
        ∀ (c : G) (s), μ (c • s) = μ s,
        ∀ c : G, Measure.map (c • ·) μ = μ,
        ∀ c : G, MeasurePreserving (c • ·) μ μ] 
Arguments: (G : Type u) {α : Type w} {m : MeasurableSpace α} [AddGroup G] [AddAction G α] [MeasurableSpace G] [MeasurableVAdd G α] (μ : MeasureTheory.Measure α)
Goal: [MeasureTheory.VAddInvariantMeasure G α μ, ∀ (c : G) (s : Set α), MeasurableSet s → μ ((fun x => c +ᵥ x) ⁻¹' s) = μ s,
    ∀ (c : G) (s : Set α), MeasurableSet s → μ (c +ᵥ s) = μ s, ∀ (c : G) (s : Set α), μ ((fun x => c +ᵥ x) ⁻¹' s) = μ s,
    ∀ (c : G) (s : Set α), μ (c +ᵥ s) = μ s, ∀ (c : G), MeasureTheory.Measure.map (fun x => c +ᵥ x) μ = μ,
    ∀ (c : G), MeasureTheory.MeasurePreserving (fun x => c +ᵥ x) μ μ].TFAE
-- MeasureTheory.smulInvariantMeasure_tfae
@[to_additive]
theorem smulInvariantMeasure_tfae :
    List.TFAE
      [SMulInvariantMeasure G α μ,
        ∀ (c : G) (s), MeasurableSet s → μ ((c • ·) ⁻¹' s) = μ s,
        ∀ (c : G) (s), MeasurableSet s → μ (c • s) = μ s,
        ∀ (c : G) (s), μ ((c • ·) ⁻¹' s) = μ s,
        ∀ (c : G) (s), μ (c • s) = μ s,
        ∀ c : G, Measure.map (c • ·) μ = μ,
        ∀ c : G, MeasurePreserving (c • ·) μ μ] 
Arguments: (G : Type u) {α : Type w} {m : MeasurableSpace α} [Group G] [MulAction G α] [MeasurableSpace G] [MeasurableSMul G α] (μ : MeasureTheory.Measure α)
Goal: [MeasureTheory.SMulInvariantMeasure G α μ, ∀ (c : G) (s : Set α), MeasurableSet s → μ ((fun x => c • x) ⁻¹' s) = μ s,
    ∀ (c : G) (s : Set α), MeasurableSet s → μ (c • s) = μ s, ∀ (c : G) (s : Set α), μ ((fun x => c • x) ⁻¹' s) = μ s,
    ∀ (c : G) (s : Set α), μ (c • s) = μ s, ∀ (c : G), MeasureTheory.Measure.map (fun x => c • x) μ = μ,
    ∀ (c : G), MeasureTheory.MeasurePreserving (fun x => c • x) μ μ].TFAE
-- MeasureTheory.NullMeasurableSet.smul
@[to_additive]
theorem NullMeasurableSet.smul {s} (hs : NullMeasurableSet s μ) (c : G) :
    NullMeasurableSet (c • s) μ 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [Group G] [MulAction G α] [MeasurableSpace G] [MeasurableSMul G α] {μ : MeasureTheory.Measure α} [MeasureTheory.SMulInvariantMeasure G α μ] {s : Set α} (hs : MeasureTheory.NullMeasurableSet s μ) (c : G)
Goal: MeasureTheory.NullMeasurableSet (c • s) μ
-- MeasureTheory.NullMeasurableSet.vadd
@[to_additive]
theorem NullMeasurableSet.smul {s} (hs : NullMeasurableSet s μ) (c : G) :
    NullMeasurableSet (c • s) μ 
Arguments: {G : Type u} {α : Type w} {m : MeasurableSpace α} [AddGroup G] [AddAction G α] [MeasurableSpace G] [MeasurableVAdd G α] {μ : MeasureTheory.Measure α} [MeasureTheory.VAddInvariantMeasure G α μ] {s : Set α} (hs : MeasureTheory.NullMeasurableSet s μ) (c : G)
Goal: MeasureTheory.NullMeasurableSet (c +ᵥ s) μ
-- MeasureTheory.measure_isOpen_pos_of_vaddInvariant_of_compact_ne_zero
@[to_additive]
theorem measure_isOpen_pos_of_smulInvariant_of_compact_ne_zero (hK : IsCompact K) (hμK : μ K ≠ 0)
    (hU : IsOpen U) (hne : U.Nonempty) : 0 < μ U 
Arguments: (G : Type u) {α : Type w} {m : MeasurableSpace α} [AddGroup G] [AddAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.VAddInvariantMeasure G α μ] [TopologicalSpace α] [ContinuousConstVAdd G α] [AddAction.IsMinimal G α] {K U : Set α} (hK : IsCompact K) (hμK : μ K ≠ 0) (hU : IsOpen U) (hne : U.Nonempty)
Goal: 0 < μ U