ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.tendsto_measure_iInter'
theorem tendsto_measure_iInter' {α ι : Type*} [MeasurableSpace α] {μ : Measure α} [Countable ι]
    [Preorder ι] [IsDirected ι (· ≤ ·)] {f : ι → Set α} (hm : ∀ i, MeasurableSet (f i))
    (hf : ∃ i, μ (f i) ≠ ∞) :
    Tendsto (fun i ↦ μ (⋂ j ∈ {j | j ≤ i}, f j)) atTop (𝓝 (μ (⋂ i, f i))) 
Arguments: {α : Type u_8} {ι : Type u_9} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [Countable ι] [Preorder ι] [IsDirected ι fun x x_1 => x ≤ x_1] {f : ι → Set α} (hm : ∀ (i : ι), MeasurableSet (f i)) (hf : ∃ i, μ (f i) ≠ ⊤)
Goal: Filter.Tendsto (fun i => μ (⋂ j ∈ {j | j ≤ i}, f j)) Filter.atTop (nhds (μ (⋂ i, f i)))
-- MeasureTheory.tendsto_measure_biInter_gt
theorem tendsto_measure_biInter_gt {ι : Type*} [LinearOrder ι] [TopologicalSpace ι]
    [OrderTopology ι] [DenselyOrdered ι] [FirstCountableTopology ι] {s : ι → Set α}
    {a : ι} (hs : ∀ r > a, MeasurableSet (s r)) (hm : ∀ i j, a < i → i ≤ j → s i ⊆ s j)
    (hf : ∃ r > a, μ (s r) ≠ ∞) : Tendsto (μ ∘ s) (𝓝[Ioi a] a) (𝓝 (μ (⋂ r > a, s r))) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_8} [LinearOrder ι] [TopologicalSpace ι] [OrderTopology ι] [DenselyOrdered ι] [FirstCountableTopology ι] {s : ι → Set α} {a : ι} (hs : ∀ r > a, MeasurableSet (s r)) (hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j) (hf : ∃ r > a, μ (s r) ≠ ⊤)
Goal: Filter.Tendsto (⇑μ ∘ s) (nhdsWithin a (Set.Ioi a)) (nhds (μ (⋂ r, ⋂ (_ : r > a), s r)))
-- MeasureTheory.measure_limsup_eq_zero
theorem measure_limsup_eq_zero {s : ℕ → Set α} (hs : (∑' i, μ (s i)) ≠ ∞) :
    μ (limsup s atTop) = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : ℕ → Set α} (hs : ∑' (i : ℕ), μ (s i) ≠ ⊤)
Goal: μ (Filter.limsup s Filter.atTop) = 0
-- MeasureTheory.measure_liminf_eq_zero
theorem measure_liminf_eq_zero {s : ℕ → Set α} (h : (∑' i, μ (s i)) ≠ ∞) :
    μ (liminf s atTop) = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : ℕ → Set α} (h : ∑' (i : ℕ), μ (s i) ≠ ⊤)
Goal: μ (Filter.liminf s Filter.atTop) = 0
-- MeasureTheory.limsup_ae_eq_of_forall_ae_eq
theorem limsup_ae_eq_of_forall_ae_eq (s : ℕ → Set α) {t : Set α}
    (h : ∀ n, s n =ᵐ[μ] t) : limsup (α := Set α) s atTop =ᵐ[μ] t 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : ℕ → Set α) {t : Set α} (h : ∀ (n : ℕ), s n =ᵐ[μ] t)
Goal: Filter.limsup s Filter.atTop =ᵐ[μ] t
-- MeasureTheory.liminf_ae_eq_of_forall_ae_eq
theorem liminf_ae_eq_of_forall_ae_eq (s : ℕ → Set α) {t : Set α}
    (h : ∀ n, s n =ᵐ[μ] t) : liminf (α := Set α) s atTop =ᵐ[μ] t 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : ℕ → Set α) {t : Set α} (h : ∀ (n : ℕ), s n =ᵐ[μ] t)
Goal: Filter.liminf s Filter.atTop =ᵐ[μ] t
-- MeasureTheory.measure_if
theorem measure_if {x : β} {t : Set β} {s : Set α} :
    μ (if x ∈ t then s else ∅) = indicator t (fun _ => μ s) x 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {x : β} {t : Set β} {s : Set α}
Goal: μ (if x ∈ t then s else ∅) = t.indicator (fun x => μ s) x
-- MeasureTheory.le_toOuterMeasure_caratheodory
theorem le_toOuterMeasure_caratheodory (μ : Measure α) : ms ≤ μ.toOuterMeasure.caratheodory 
Arguments: {α : Type u_1} [ms : MeasurableSpace α] (μ : MeasureTheory.Measure α)
Goal: ms ≤ μ.caratheodory
-- MeasureTheory.toMeasure_toOuterMeasure
@[simp]
theorem toMeasure_toOuterMeasure (m : OuterMeasure α) (h : ms ≤ m.caratheodory) :
    (m.toMeasure h).toOuterMeasure = m.trim 
Arguments: {α : Type u_1} [ms : MeasurableSpace α] (m : MeasureTheory.OuterMeasure α) (h : ms ≤ m.caratheodory)
Goal: (m.toMeasure h).toOuterMeasure = m.trim
-- MeasureTheory.toMeasure_apply
@[simp]
theorem toMeasure_apply (m : OuterMeasure α) (h : ms ≤ m.caratheodory) {s : Set α}
    (hs : MeasurableSet s) : m.toMeasure h s = m s 
Arguments: {α : Type u_1} [ms : MeasurableSpace α] (m : MeasureTheory.OuterMeasure α) (h : ms ≤ m.caratheodory) {s : Set α} (hs : MeasurableSet s)
Goal: (m.toMeasure h) s = m s
-- MeasureTheory.le_toMeasure_apply
theorem le_toMeasure_apply (m : OuterMeasure α) (h : ms ≤ m.caratheodory) (s : Set α) :
    m s ≤ m.toMeasure h s 
Arguments: {α : Type u_1} [ms : MeasurableSpace α] (m : MeasureTheory.OuterMeasure α) (h : ms ≤ m.caratheodory) (s : Set α)
Goal: m s ≤ (m.toMeasure h) s
-- MeasureTheory.toMeasure_apply₀
theorem toMeasure_apply₀ (m : OuterMeasure α) (h : ms ≤ m.caratheodory) {s : Set α}
    (hs : NullMeasurableSet s (m.toMeasure h)) : m.toMeasure h s = m s 
Arguments: {α : Type u_1} [ms : MeasurableSpace α] (m : MeasureTheory.OuterMeasure α) (h : ms ≤ m.caratheodory) {s : Set α} (hs : MeasureTheory.NullMeasurableSet s (m.toMeasure h))
Goal: (m.toMeasure h) s = m s
-- MeasureTheory.toOuterMeasure_toMeasure
@[simp]
theorem toOuterMeasure_toMeasure {μ : Measure α} :
    μ.toOuterMeasure.toMeasure (le_toOuterMeasure_caratheodory _) = μ 
Arguments: {α : Type u_1} [ms : MeasurableSpace α] {μ : MeasureTheory.Measure α}
Goal: μ.toMeasure ⋯ = μ
-- MeasureTheory.boundedBy_measure
@[simp]
theorem boundedBy_measure (μ : Measure α) : OuterMeasure.boundedBy μ = μ.toOuterMeasure 
Arguments: {α : Type u_1} [ms : MeasurableSpace α] (μ : MeasureTheory.Measure α)
Goal: MeasureTheory.OuterMeasure.boundedBy ⇑μ = μ.toOuterMeasure
-- MeasureTheory.Measure.measure_inter_eq_of_measure_eq
theorem measure_inter_eq_of_measure_eq {s t u : Set α} (hs : MeasurableSet s) (h : μ t = μ u)
    (htu : t ⊆ u) (ht_ne_top : μ t ≠ ∞) : μ (t ∩ s) = μ (u ∩ s) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t u : Set α} (hs : MeasurableSet s) (h : μ t = μ u) (htu : t ⊆ u) (ht_ne_top : μ t ≠ ⊤)
Goal: μ (t ∩ s) = μ (u ∩ s)
-- MeasureTheory.Measure.measure_toMeasurable_inter
theorem measure_toMeasurable_inter {s t : Set α} (hs : MeasurableSet s) (ht : μ t ≠ ∞) :
    μ (toMeasurable μ t ∩ s) = μ (t ∩ s) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} (hs : MeasurableSet s) (ht : μ t ≠ ⊤)
Goal: μ (MeasureTheory.toMeasurable μ t ∩ s) = μ (t ∩ s)
-- MeasureTheory.Measure.zero_toOuterMeasure
@[simp]
theorem zero_toOuterMeasure {_m : MeasurableSpace α} : (0 : Measure α).toOuterMeasure = 0 
Arguments: {α : Type u_1} {_m : MeasurableSpace α}
Goal: MeasureTheory.Measure.toOuterMeasure 0 = 0
-- MeasureTheory.Measure.coe_zero
@[simp, norm_cast]
theorem coe_zero {_m : MeasurableSpace α} : ⇑(0 : Measure α) = 0 
Arguments: {α : Type u_1} {_m : MeasurableSpace α}
Goal: ⇑0 = 0
-- MeasureTheory.Measure.apply_eq_zero_of_isEmpty
@[nontriviality]
lemma apply_eq_zero_of_isEmpty [IsEmpty α] {_ : MeasurableSpace α} (μ : Measure α) (s : Set α) :
    μ s = 0 
Arguments: {α : Type u_1} [IsEmpty α]
Goal: ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α) (s : Set α), μ s = 0
-- MeasureTheory.Measure.eq_zero_of_isEmpty
theorem eq_zero_of_isEmpty [IsEmpty α] {_m : MeasurableSpace α} (μ : Measure α) : μ = 0 
Arguments: {α : Type u_1} [IsEmpty α] {_m : MeasurableSpace α} (μ : MeasureTheory.Measure α)
Goal: μ = 0
-- MeasureTheory.Measure.add_toOuterMeasure
@[simp]
theorem add_toOuterMeasure {_m : MeasurableSpace α} (μ₁ μ₂ : Measure α) :
    (μ₁ + μ₂).toOuterMeasure = μ₁.toOuterMeasure + μ₂.toOuterMeasure 
Arguments: {α : Type u_1} {_m : MeasurableSpace α} (μ₁ μ₂ : MeasureTheory.Measure α)
Goal: (μ₁ + μ₂).toOuterMeasure = μ₁.toOuterMeasure + μ₂.toOuterMeasure
-- MeasureTheory.Measure.coe_add
@[simp, norm_cast]
theorem coe_add {_m : MeasurableSpace α} (μ₁ μ₂ : Measure α) : ⇑(μ₁ + μ₂) = μ₁ + μ₂ 
Arguments: {α : Type u_1} {_m : MeasurableSpace α} (μ₁ μ₂ : MeasureTheory.Measure α)
Goal: ⇑(μ₁ + μ₂) = ⇑μ₁ + ⇑μ₂
-- MeasureTheory.Measure.add_apply
theorem add_apply {_m : MeasurableSpace α} (μ₁ μ₂ : Measure α) (s : Set α) :
    (μ₁ + μ₂) s = μ₁ s + μ₂ s 
Arguments: {α : Type u_1} {_m : MeasurableSpace α} (μ₁ μ₂ : MeasureTheory.Measure α) (s : Set α)
Goal: (μ₁ + μ₂) s = μ₁ s + μ₂ s
-- MeasureTheory.Measure.smul_toOuterMeasure
@[simp]
theorem smul_toOuterMeasure {_m : MeasurableSpace α} (c : R) (μ : Measure α) :
    (c • μ).toOuterMeasure = c • μ.toOuterMeasure 
Arguments: {α : Type u_1} {R : Type u_6} [SMul R ENNReal] [IsScalarTower R ENNReal ENNReal] {_m : MeasurableSpace α} (c : R) (μ : MeasureTheory.Measure α)
Goal: (c • μ).toOuterMeasure = c • μ.toOuterMeasure
-- MeasureTheory.Measure.coe_smul
@[simp, norm_cast]
theorem coe_smul {_m : MeasurableSpace α} (c : R) (μ : Measure α) : ⇑(c • μ) = c • ⇑μ 
Arguments: {α : Type u_1} {R : Type u_6} [SMul R ENNReal] [IsScalarTower R ENNReal ENNReal] {_m : MeasurableSpace α} (c : R) (μ : MeasureTheory.Measure α)
Goal: ⇑(c • μ) = c • ⇑μ
-- MeasureTheory.Measure.smul_apply
@[simp]
theorem smul_apply {_m : MeasurableSpace α} (c : R) (μ : Measure α) (s : Set α) :
    (c • μ) s = c • μ s 
Arguments: {α : Type u_1} {R : Type u_6} [SMul R ENNReal] [IsScalarTower R ENNReal ENNReal] {_m : MeasurableSpace α} (c : R) (μ : MeasureTheory.Measure α) (s : Set α)
Goal: (c • μ) s = c • μ s
-- MeasureTheory.Measure.coe_finset_sum
@[simp]
theorem coe_finset_sum {_m : MeasurableSpace α} (I : Finset ι) (μ : ι → Measure α) :
    ⇑(∑ i ∈ I, μ i) = ∑ i ∈ I, ⇑(μ i) 
Arguments: {α : Type u_1} {ι : Type u_5} {_m : MeasurableSpace α} (I : Finset ι) (μ : ι → MeasureTheory.Measure α)
Goal: ⇑(∑ i ∈ I, μ i) = ∑ i ∈ I, ⇑(μ i)
-- MeasureTheory.Measure.finset_sum_apply
theorem finset_sum_apply {m : MeasurableSpace α} (I : Finset ι) (μ : ι → Measure α) (s : Set α) :
    (∑ i ∈ I, μ i) s = ∑ i ∈ I, μ i s 
Arguments: {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} (I : Finset ι) (μ : ι → MeasureTheory.Measure α) (s : Set α)
Goal: (∑ i ∈ I, μ i) s = ∑ i ∈ I, (μ i) s
-- MeasureTheory.Measure.coe_nnreal_smul_apply
@[simp]
theorem coe_nnreal_smul_apply {_m : MeasurableSpace α} (c : ℝ≥0) (μ : Measure α) (s : Set α) :
    (c • μ) s = c * μ s 
Arguments: {α : Type u_1} {_m : MeasurableSpace α} (c : NNReal) (μ : MeasureTheory.Measure α) (s : Set α)
Goal: (c • μ) s = ↑c * μ s
-- MeasureTheory.Measure.nnreal_smul_coe_apply
@[simp]
theorem nnreal_smul_coe_apply {_m : MeasurableSpace α} (c : ℝ≥0) (μ : Measure α) (s : Set α) :
    c • μ s = c * μ s 
Arguments: {α : Type u_1} {_m : MeasurableSpace α} (c : NNReal) (μ : MeasureTheory.Measure α) (s : Set α)
Goal: c • μ s = ↑c * μ s
-- MeasureTheory.Measure.ae_smul_measure_iff
theorem ae_smul_measure_iff {p : α → Prop} {c : ℝ≥0∞} (hc : c ≠ 0) :
    (∀ᵐ x ∂c • μ, p x) ↔ ∀ᵐ x ∂μ, p x 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : α → Prop} {c : ENNReal} (hc : c ≠ 0)
Goal: (∀ᵐ (x : α) ∂c • μ, p x) ↔ ∀ᵐ (x : α) ∂μ, p x
-- MeasureTheory.Measure.ae_smul_measure_eq
@[simp]
theorem ae_smul_measure_eq {c : ℝ≥0∞} (hc : c ≠ 0) : ae (c • μ) = ae μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {c : ENNReal} (hc : c ≠ 0)
Goal: MeasureTheory.ae (c • μ) = MeasureTheory.ae μ
-- MeasureTheory.Measure.measure_eq_left_of_subset_of_measure_add_eq
theorem measure_eq_left_of_subset_of_measure_add_eq {s t : Set α} (h : (μ + ν) t ≠ ∞) (h' : s ⊆ t)
    (h'' : (μ + ν) s = (μ + ν) t) : μ s = μ t 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {s t : Set α} (h : (μ + ν) t ≠ ⊤) (h' : s ⊆ t) (h'' : (μ + ν) s = (μ + ν) t)
Goal: μ s = μ t
-- MeasureTheory.Measure.measure_eq_right_of_subset_of_measure_add_eq
theorem measure_eq_right_of_subset_of_measure_add_eq {s t : Set α} (h : (μ + ν) t ≠ ∞) (h' : s ⊆ t)
    (h'' : (μ + ν) s = (μ + ν) t) : ν s = ν t 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {s t : Set α} (h : (μ + ν) t ≠ ⊤) (h' : s ⊆ t) (h'' : (μ + ν) s = (μ + ν) t)
Goal: ν s = ν t
-- MeasureTheory.Measure.measure_toMeasurable_add_inter_left
theorem measure_toMeasurable_add_inter_left {s t : Set α} (hs : MeasurableSet s)
    (ht : (μ + ν) t ≠ ∞) : μ (toMeasurable (μ + ν) t ∩ s) = μ (t ∩ s) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {s t : Set α} (hs : MeasurableSet s) (ht : (μ + ν) t ≠ ⊤)
Goal: μ (MeasureTheory.toMeasurable (μ + ν) t ∩ s) = μ (t ∩ s)
-- MeasureTheory.Measure.measure_toMeasurable_add_inter_right
theorem measure_toMeasurable_add_inter_right {s t : Set α} (hs : MeasurableSet s)
    (ht : (μ + ν) t ≠ ∞) : ν (toMeasurable (μ + ν) t ∩ s) = ν (t ∩ s) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {s t : Set α} (hs : MeasurableSet s) (ht : (μ + ν) t ≠ ⊤)
Goal: ν (MeasureTheory.toMeasurable (μ + ν) t ∩ s) = ν (t ∩ s)
-- MeasureTheory.Measure.toOuterMeasure_le
theorem toOuterMeasure_le : μ₁.toOuterMeasure ≤ μ₂.toOuterMeasure ↔ μ₁ ≤ μ₂ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ₁ μ₂ : MeasureTheory.Measure α}
Goal: μ₁.toOuterMeasure ≤ μ₂.toOuterMeasure ↔ μ₁ ≤ μ₂
-- MeasureTheory.Measure.le_iff
theorem le_iff : μ₁ ≤ μ₂ ↔ ∀ s, MeasurableSet s → μ₁ s ≤ μ₂ s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ₁ μ₂ : MeasureTheory.Measure α}
Goal: μ₁ ≤ μ₂ ↔ ∀ (s : Set α), MeasurableSet s → μ₁ s ≤ μ₂ s
-- MeasureTheory.Measure.le_intro
theorem le_intro (h : ∀ s, MeasurableSet s → s.Nonempty → μ₁ s ≤ μ₂ s) : μ₁ ≤ μ₂ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ₁ μ₂ : MeasureTheory.Measure α} (h : ∀ (s : Set α), MeasurableSet s → s.Nonempty → μ₁ s ≤ μ₂ s)
Goal: μ₁ ≤ μ₂
-- MeasureTheory.Measure.le_iff'
theorem le_iff' : μ₁ ≤ μ₂ ↔ ∀ s, μ₁ s ≤ μ₂ s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ₁ μ₂ : MeasureTheory.Measure α}
Goal: μ₁ ≤ μ₂ ↔ ∀ (s : Set α), μ₁ s ≤ μ₂ s
-- MeasureTheory.Measure.lt_iff
theorem lt_iff : μ < ν ↔ μ ≤ ν ∧ ∃ s, MeasurableSet s ∧ μ s < ν s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α}
Goal: μ < ν ↔ μ ≤ ν ∧ ∃ s, MeasurableSet s ∧ μ s < ν s
-- MeasureTheory.Measure.lt_iff'
theorem lt_iff' : μ < ν ↔ μ ≤ ν ∧ ∃ s, μ s < ν s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α}
Goal: μ < ν ↔ μ ≤ ν ∧ ∃ s, μ s < ν s
-- MeasureTheory.Measure.le_add_left
protected theorem le_add_left (h : μ ≤ ν) : μ ≤ ν' + ν 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν ν' : MeasureTheory.Measure α} (h : μ ≤ ν)
Goal: μ ≤ ν' + ν
-- MeasureTheory.Measure.le_add_right
protected theorem le_add_right (h : μ ≤ ν) : μ ≤ ν + ν' 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν ν' : MeasureTheory.Measure α} (h : μ ≤ ν)
Goal: μ ≤ ν + ν'
-- MeasureTheory.Measure.sInf_caratheodory
theorem sInf_caratheodory (s : Set α) (hs : MeasurableSet s) :
    MeasurableSet[(sInf (toOuterMeasure '' m)).caratheodory] s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {m : Set (MeasureTheory.Measure α)} (s : Set α) (hs : MeasurableSet s)
Goal: MeasurableSet s
-- MeasureTheory.Measure.sInf_apply
theorem sInf_apply (hs : MeasurableSet s) : sInf m s = sInf (toOuterMeasure '' m) s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {s : Set α} {m : Set (MeasureTheory.Measure α)} (hs : MeasurableSet s)
Goal: (sInf m) s = (sInf (MeasureTheory.Measure.toOuterMeasure '' m)) s
-- _private.Mathlib.MeasureTheory.Measure.MeasureSpace.0.MeasureTheory.Measure.measure_sInf_le
private theorem measure_sInf_le (h : μ ∈ m) : sInf m ≤ μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {m : Set (MeasureTheory.Measure α)} (h : μ ∈ m)
Goal: sInf m ≤ μ
-- _private.Mathlib.MeasureTheory.Measure.MeasureSpace.0.MeasureTheory.Measure.measure_le_sInf
private theorem measure_le_sInf (h : ∀ μ' ∈ m, μ ≤ μ') : μ ≤ sInf m 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {m : Set (MeasureTheory.Measure α)} (h : ∀ μ' ∈ m, μ ≤ μ')
Goal: μ ≤ sInf m
-- MeasureTheory.OuterMeasure.toMeasure_top
@[simp]
theorem _root_.MeasureTheory.OuterMeasure.toMeasure_top :
    (⊤ : OuterMeasure α).toMeasure (by rw [OuterMeasure.top_caratheodory]; exact le_top) =
      (⊤ : Measure α) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α}
Goal: ⊤.toMeasure ⋯ = ⊤
-- MeasureTheory.Measure.toOuterMeasure_top
@[simp]
theorem toOuterMeasure_top [MeasurableSpace α] :
    (⊤ : Measure α).toOuterMeasure = (⊤ : OuterMeasure α) 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: ⊤.toOuterMeasure = ⊤
-- MeasureTheory.Measure.top_add
@[simp]
theorem top_add : ⊤ + μ = ⊤ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: ⊤ + μ = ⊤
-- MeasureTheory.Measure.add_top
@[simp]
theorem add_top : μ + ⊤ = ⊤ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: μ + ⊤ = ⊤
-- MeasureTheory.Measure.zero_le
protected theorem zero_le {_m0 : MeasurableSpace α} (μ : Measure α) : 0 ≤ μ 
Arguments: {α : Type u_1} {_m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α)
Goal: 0 ≤ μ
-- MeasureTheory.Measure.nonpos_iff_eq_zero'
theorem nonpos_iff_eq_zero' : μ ≤ 0 ↔ μ = 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: μ ≤ 0 ↔ μ = 0
-- MeasureTheory.Measure.measure_univ_eq_zero
@[simp]
theorem measure_univ_eq_zero : μ univ = 0 ↔ μ = 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: μ Set.univ = 0 ↔ μ = 0
-- MeasureTheory.Measure.measure_univ_ne_zero
theorem measure_univ_ne_zero : μ univ ≠ 0 ↔ μ ≠ 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: μ Set.univ ≠ 0 ↔ μ ≠ 0
-- MeasureTheory.Measure.measure_univ_pos
@[simp]
theorem measure_univ_pos : 0 < μ univ ↔ μ ≠ 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: 0 < μ Set.univ ↔ μ ≠ 0
-- MeasureTheory.Measure.liftLinear_apply₀
lemma liftLinear_apply₀ {f : OuterMeasure α →ₗ[ℝ≥0∞] OuterMeasure β} (hf) {s : Set β}
    (hs : NullMeasurableSet s (liftLinear f hf μ)) : liftLinear f hf μ s = f μ.toOuterMeasure s 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : MeasureTheory.OuterMeasure α →ₗ[ENNReal] MeasureTheory.OuterMeasure β} (hf : ∀ (μ : MeasureTheory.Measure α), inst✝ ≤ (f μ.toOuterMeasure).caratheodory) {s : Set β} (hs : MeasureTheory.NullMeasurableSet s ((MeasureTheory.Measure.liftLinear f hf) μ))
Goal: ((MeasureTheory.Measure.liftLinear f hf) μ) s = (f μ.toOuterMeasure) s
-- MeasureTheory.Measure.liftLinear_apply
@[simp]
theorem liftLinear_apply {f : OuterMeasure α →ₗ[ℝ≥0∞] OuterMeasure β} (hf) {s : Set β}
    (hs : MeasurableSet s) : liftLinear f hf μ s = f μ.toOuterMeasure s 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : MeasureTheory.OuterMeasure α →ₗ[ENNReal] MeasureTheory.OuterMeasure β} (hf : ∀ (μ : MeasureTheory.Measure α), inst✝ ≤ (f μ.toOuterMeasure).caratheodory) {s : Set β} (hs : MeasurableSet s)
Goal: ((MeasureTheory.Measure.liftLinear f hf) μ) s = (f μ.toOuterMeasure) s
-- MeasureTheory.Measure.le_liftLinear_apply
theorem le_liftLinear_apply {f : OuterMeasure α →ₗ[ℝ≥0∞] OuterMeasure β} (hf) (s : Set β) :
    f μ.toOuterMeasure s ≤ liftLinear f hf μ s 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : MeasureTheory.OuterMeasure α →ₗ[ENNReal] MeasureTheory.OuterMeasure β} (hf : ∀ (μ : MeasureTheory.Measure α), inst✝ ≤ (f μ.toOuterMeasure).caratheodory) (s : Set β)
Goal: (f μ.toOuterMeasure) s ≤ ((MeasureTheory.Measure.liftLinear f hf) μ) s
-- MeasureTheory.Measure.mapₗ_congr
theorem mapₗ_congr {f g : α → β} (hf : Measurable f) (hg : Measurable g) (h : f =ᵐ[μ] g) :
    mapₗ f μ = mapₗ g μ 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {f g : α → β} (hf : Measurable f) (hg : Measurable g) (h : f =ᵐ[μ] g)
Goal: (MeasureTheory.Measure.mapₗ f) μ = (MeasureTheory.Measure.mapₗ g) μ
-- MeasureTheory.Measure.map_def
irreducible_def map [MeasurableSpace α] (f : α → β) (μ : Measure α) : Measure β 
Arguments: {α : Type u_8} {β : Type u_9} [MeasurableSpace β] [MeasurableSpace α] (f : α → β) (μ : MeasureTheory.Measure α)
Goal: MeasureTheory.Measure.map f μ =
  if hf : AEMeasurable f μ then (MeasureTheory.Measure.mapₗ (AEMeasurable.mk f hf)) μ else 0
-- MeasureTheory.Measure.mapₗ_mk_apply_of_aemeasurable
theorem mapₗ_mk_apply_of_aemeasurable {f : α → β} (hf : AEMeasurable f μ) :
    mapₗ (hf.mk f) μ = map f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β} (hf : AEMeasurable f μ)
Goal: (MeasureTheory.Measure.mapₗ (AEMeasurable.mk f hf)) μ = MeasureTheory.Measure.map f μ
-- MeasureTheory.Measure.mapₗ_apply_of_measurable
theorem mapₗ_apply_of_measurable {f : α → β} (hf : Measurable f) (μ : Measure α) :
    mapₗ f μ = map f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} (hf : Measurable f) (μ : MeasureTheory.Measure α)
Goal: (MeasureTheory.Measure.mapₗ f) μ = MeasureTheory.Measure.map f μ
-- MeasureTheory.Measure.map_add
@[simp]
theorem map_add (μ ν : Measure α) {f : α → β} (hf : Measurable f) :
    (μ + ν).map f = μ.map f + ν.map f 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] (μ ν : MeasureTheory.Measure α) {f : α → β} (hf : Measurable f)
Goal: MeasureTheory.Measure.map f (μ + ν) = MeasureTheory.Measure.map f μ + MeasureTheory.Measure.map f ν
-- MeasureTheory.Measure.map_zero
@[simp]
theorem map_zero (f : α → β) : (0 : Measure α).map f = 0 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] (f : α → β)
Goal: MeasureTheory.Measure.map f 0 = 0
-- MeasureTheory.Measure.map_of_not_aemeasurable
@[simp]
theorem map_of_not_aemeasurable {f : α → β} {μ : Measure α} (hf : ¬AEMeasurable f μ) :
    μ.map f = 0 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {f : α → β} {μ : MeasureTheory.Measure α} (hf : ¬AEMeasurable f μ)
Goal: MeasureTheory.Measure.map f μ = 0
-- MeasureTheory.Measure.map_congr
theorem map_congr {f g : α → β} (h : f =ᵐ[μ] g) : Measure.map f μ = Measure.map g μ 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {f g : α → β} (h : f =ᵐ[μ] g)
Goal: MeasureTheory.Measure.map f μ = MeasureTheory.Measure.map g μ
-- MeasureTheory.Measure.map_smul
@[simp]
protected theorem map_smul (c : ℝ≥0∞) (μ : Measure α) (f : α → β) :
    (c • μ).map f = c • μ.map f 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] (c : ENNReal) (μ : MeasureTheory.Measure α) (f : α → β)
Goal: MeasureTheory.Measure.map f (c • μ) = c • MeasureTheory.Measure.map f μ
-- MeasureTheory.Measure.map_smul_nnreal
@[simp]
protected theorem map_smul_nnreal (c : ℝ≥0) (μ : Measure α) (f : α → β) :
    (c • μ).map f = c • μ.map f 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] (c : NNReal) (μ : MeasureTheory.Measure α) (f : α → β)
Goal: MeasureTheory.Measure.map f (c • μ) = c • MeasureTheory.Measure.map f μ
-- MeasureTheory.Measure.map_apply₀
lemma map_apply₀ {f : α → β} (hf : AEMeasurable f μ) {s : Set β}
    (hs : NullMeasurableSet s (map f μ)) : μ.map f s = μ (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β} (hf : AEMeasurable f μ) {s : Set β} (hs : MeasureTheory.NullMeasurableSet s (MeasureTheory.Measure.map f μ))
Goal: (MeasureTheory.Measure.map f μ) s = μ (f ⁻¹' s)
-- MeasureTheory.Measure.map_apply_of_aemeasurable
@[simp]
theorem map_apply_of_aemeasurable (hf : AEMeasurable f μ) {s : Set β} (hs : MeasurableSet s) :
    μ.map f s = μ (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β} (hf : AEMeasurable f μ) {s : Set β} (hs : MeasurableSet s)
Goal: (MeasureTheory.Measure.map f μ) s = μ (f ⁻¹' s)
-- MeasureTheory.Measure.map_apply
@[simp]
theorem map_apply (hf : Measurable f) {s : Set β} (hs : MeasurableSet s) :
    μ.map f s = μ (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β} (hf : Measurable f) {s : Set β} (hs : MeasurableSet s)
Goal: (MeasureTheory.Measure.map f μ) s = μ (f ⁻¹' s)