ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.ae_restrict_mem
theorem ae_restrict_mem (hs : MeasurableSet s) : ∀ᵐ x ∂μ.restrict s, x ∈ s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s)
Goal: ∀ᵐ (x : α) ∂μ.restrict s, x ∈ s
-- MeasureTheory.ae_restrict_of_ae
theorem ae_restrict_of_ae {s : Set α} {p : α → Prop} (h : ∀ᵐ x ∂μ, p x) : ∀ᵐ x ∂μ.restrict s, p x 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {p : α → Prop} (h : ∀ᵐ (x : α) ∂μ, p x)
Goal: ∀ᵐ (x : α) ∂μ.restrict s, p x
-- MeasureTheory.ae_restrict_of_ae_restrict_of_subset
theorem ae_restrict_of_ae_restrict_of_subset {s t : Set α} {p : α → Prop} (hst : s ⊆ t)
    (h : ∀ᵐ x ∂μ.restrict t, p x) : ∀ᵐ x ∂μ.restrict s, p x 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} {p : α → Prop} (hst : s ⊆ t) (h : ∀ᵐ (x : α) ∂μ.restrict t, p x)
Goal: ∀ᵐ (x : α) ∂μ.restrict s, p x
-- MeasureTheory.ae_of_ae_restrict_of_ae_restrict_compl
theorem ae_of_ae_restrict_of_ae_restrict_compl (t : Set α) {p : α → Prop}
    (ht : ∀ᵐ x ∂μ.restrict t, p x) (htc : ∀ᵐ x ∂μ.restrict tᶜ, p x) : ∀ᵐ x ∂μ, p x 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (t : Set α) {p : α → Prop} (ht : ∀ᵐ (x : α) ∂μ.restrict t, p x) (htc : ∀ᵐ (x : α) ∂μ.restrict tᶜ, p x)
Goal: ∀ᵐ (x : α) ∂μ, p x
-- MeasureTheory.mem_map_restrict_ae_iff
theorem mem_map_restrict_ae_iff {β} {s : Set α} {t : Set β} {f : α → β} (hs : MeasurableSet s) :
    t ∈ Filter.map f (ae (μ.restrict s)) ↔ μ ((f ⁻¹' t)ᶜ ∩ s) = 0 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} {s : Set α} {t : Set β} {f : α → β} (hs : MeasurableSet s)
Goal: t ∈ Filter.map f (MeasureTheory.ae (μ.restrict s)) ↔ μ ((f ⁻¹' t)ᶜ ∩ s) = 0
-- MeasureTheory.ae_smul_measure
theorem ae_smul_measure {p : α → Prop} [Monoid R] [DistribMulAction R ℝ≥0∞]
    [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (h : ∀ᵐ x ∂μ, p x) (c : R) : ∀ᵐ x ∂c • μ, p x 
Arguments: {R : Type u_1} {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : α → Prop} [Monoid R] [DistribMulAction R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (h : ∀ᵐ (x : α) ∂μ, p x) (c : R)
Goal: ∀ᵐ (x : α) ∂c • μ, p x
-- MeasureTheory.ae_add_measure_iff
theorem ae_add_measure_iff {p : α → Prop} {ν} :
    (∀ᵐ x ∂μ + ν, p x) ↔ (∀ᵐ x ∂μ, p x) ∧ ∀ᵐ x ∂ν, p x 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : α → Prop} {ν : MeasureTheory.Measure α}
Goal: (∀ᵐ (x : α) ∂μ + ν, p x) ↔ (∀ᵐ (x : α) ∂μ, p x) ∧ ∀ᵐ (x : α) ∂ν, p x
-- MeasureTheory.ae_eq_comp'
theorem ae_eq_comp' {ν : Measure β} {f : α → β} {g g' : β → δ} (hf : AEMeasurable f μ)
    (h : g =ᵐ[ν] g') (h2 : μ.map f ≪ ν) : g ∘ f =ᵐ[μ] g' ∘ f 
Arguments: {α : Type u_2} {β : Type u_3} {δ : Type u_4} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → β} {g g' : β → δ} (hf : AEMeasurable f μ) (h : g =ᵐ[ν] g') (h2 : MeasureTheory.Measure.map f μ ≪ ν)
Goal: g ∘ f =ᵐ[μ] g' ∘ f
-- MeasureTheory.Measure.QuasiMeasurePreserving.ae_eq_comp
theorem Measure.QuasiMeasurePreserving.ae_eq_comp {ν : Measure β} {f : α → β} {g g' : β → δ}
    (hf : QuasiMeasurePreserving f μ ν) (h : g =ᵐ[ν] g') : g ∘ f =ᵐ[μ] g' ∘ f 
Arguments: {α : Type u_2} {β : Type u_3} {δ : Type u_4} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → β} {g g' : β → δ} (hf : MeasureTheory.Measure.QuasiMeasurePreserving f μ ν) (h : g =ᵐ[ν] g')
Goal: g ∘ f =ᵐ[μ] g' ∘ f
-- MeasureTheory.ae_eq_comp
theorem ae_eq_comp {f : α → β} {g g' : β → δ} (hf : AEMeasurable f μ) (h : g =ᵐ[μ.map f] g') :
    g ∘ f =ᵐ[μ] g' ∘ f 
Arguments: {α : Type u_2} {β : Type u_3} {δ : Type u_4} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β} {g g' : β → δ} (hf : AEMeasurable f μ) (h : g =ᵐ[MeasureTheory.Measure.map f μ] g')
Goal: g ∘ f =ᵐ[μ] g' ∘ f
-- MeasureTheory.div_ae_eq_one
@[to_additive]
theorem div_ae_eq_one {β} [Group β] (f g : α → β) : f / g =ᵐ[μ] 1 ↔ f =ᵐ[μ] g 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [Group β] (f g : α → β)
Goal: f / g =ᵐ[μ] 1 ↔ f =ᵐ[μ] g
-- MeasureTheory.sub_ae_eq_zero
@[to_additive]
theorem div_ae_eq_one {β} [Group β] (f g : α → β) : f / g =ᵐ[μ] 1 ↔ f =ᵐ[μ] g 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [AddGroup β] (f g : α → β)
Goal: f - g =ᵐ[μ] 0 ↔ f =ᵐ[μ] g
-- MeasureTheory.one_le_div_ae
@[to_additive sub_nonneg_ae]
lemma one_le_div_ae {β : Type*} [Group β] [LE β]
    [CovariantClass β β (Function.swap (· * ·)) (· ≤ ·)] (f g : α → β) :
    1 ≤ᵐ[μ] g / f ↔ f ≤ᵐ[μ] g 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [Group β] [LE β] [CovariantClass β β (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (f g : α → β)
Goal: 1 ≤ᵐ[μ] g / f ↔ f ≤ᵐ[μ] g
-- MeasureTheory.sub_nonneg_ae
@[to_additive sub_nonneg_ae]
lemma one_le_div_ae {β : Type*} [Group β] [LE β]
    [CovariantClass β β (Function.swap (· * ·)) (· ≤ ·)] (f g : α → β) :
    1 ≤ᵐ[μ] g / f ↔ f ≤ᵐ[μ] g 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [AddGroup β] [LE β] [CovariantClass β β (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (f g : α → β)
Goal: 0 ≤ᵐ[μ] g - f ↔ f ≤ᵐ[μ] g
-- MeasureTheory.le_ae_restrict
theorem le_ae_restrict : ae μ ⊓ 𝓟 s ≤ ae (μ.restrict s) 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α}
Goal: MeasureTheory.ae μ ⊓ 𝓟 s ≤ MeasureTheory.ae (μ.restrict s)
-- MeasureTheory.ae_restrict_eq
@[simp]
theorem ae_restrict_eq (hs : MeasurableSet s) : ae (μ.restrict s) = ae μ ⊓ 𝓟 s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s)
Goal: MeasureTheory.ae (μ.restrict s) = MeasureTheory.ae μ ⊓ 𝓟 s
-- MeasureTheory.ae_restrict_eq_bot
theorem ae_restrict_eq_bot {s} : ae (μ.restrict s) = ⊥ ↔ μ s = 0 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α}
Goal: MeasureTheory.ae (μ.restrict s) = ⊥ ↔ μ s = 0
-- MeasureTheory.ae_restrict_neBot
theorem ae_restrict_neBot {s} : (ae <| μ.restrict s).NeBot ↔ μ s ≠ 0 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α}
Goal: (MeasureTheory.ae (μ.restrict s)).NeBot ↔ μ s ≠ 0
-- MeasureTheory.self_mem_ae_restrict
theorem self_mem_ae_restrict {s} (hs : MeasurableSet s) : s ∈ ae (μ.restrict s) 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s)
Goal: s ∈ MeasureTheory.ae (μ.restrict s)
-- MeasureTheory.ae_restrict_of_ae_eq_of_ae_restrict
theorem ae_restrict_of_ae_eq_of_ae_restrict {s t} (hst : s =ᵐ[μ] t) {p : α → Prop} :
    (∀ᵐ x ∂μ.restrict s, p x) → ∀ᵐ x ∂μ.restrict t, p x 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : α → Prop} (hst : s =ᵐ[μ] t) {p : α → Prop}
Goal: (∀ᵐ (x : α) ∂μ.restrict s, p x) → ∀ᵐ (x : α) ∂μ.restrict t, p x
-- MeasureTheory.ae_restrict_congr_set
theorem ae_restrict_congr_set {s t} (hst : s =ᵐ[μ] t) {p : α → Prop} :
    (∀ᵐ x ∂μ.restrict s, p x) ↔ ∀ᵐ x ∂μ.restrict t, p x 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : α → Prop} (hst : s =ᵐ[μ] t) {p : α → Prop}
Goal: (∀ᵐ (x : α) ∂μ.restrict s, p x) ↔ ∀ᵐ (x : α) ∂μ.restrict t, p x
-- MeasureTheory.measure_setOf_frequently_eq_zero
theorem measure_setOf_frequently_eq_zero {p : ℕ → α → Prop} (hp : ∑' i, μ { x | p i x } ≠ ∞) :
    μ { x | ∃ᶠ n in atTop, p n x } = 0 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ℕ → α → Prop} (hp : ∑' (i : ℕ), μ {x | p i x} ≠ ⊤)
Goal: μ {x | ∃ᶠ (n : ℕ) in Filter.atTop, p n x} = 0
-- MeasureTheory.ae_eventually_not_mem
theorem ae_eventually_not_mem {s : ℕ → Set α} (hs : (∑' i, μ (s i)) ≠ ∞) :
    ∀ᵐ x ∂μ, ∀ᶠ n in atTop, x ∉ s n 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : ℕ → Set α} (hs : ∑' (i : ℕ), μ (s i) ≠ ⊤)
Goal: ∀ᵐ (x : α) ∂μ, ∀ᶠ (n : ℕ) in Filter.atTop, x ∉ s n
-- MeasureTheory.NullMeasurable.measure_preimage_eq_measure_restrict_preimage_of_ae_compl_eq_const
lemma NullMeasurable.measure_preimage_eq_measure_restrict_preimage_of_ae_compl_eq_const
    {β : Type*} [MeasurableSpace β] {b : β} {f : α → β} {s : Set α}
    (f_mble : NullMeasurable f (μ.restrict s)) (hs : f =ᵐ[Measure.restrict μ sᶜ] (fun _ ↦ b))
    {t : Set β} (t_mble : MeasurableSet t) (ht : b ∉ t) :
    μ (f ⁻¹' t) = μ.restrict s (f ⁻¹' t) 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [MeasurableSpace β] {b : β} {f : α → β} {s : Set α} (f_mble : MeasureTheory.NullMeasurable f (μ.restrict s)) (hs : f =ᵐ[μ.restrict sᶜ] fun x => b) {t : Set β} (t_mble : MeasurableSet t) (ht : b ∉ t)
Goal: μ (f ⁻¹' t) = (μ.restrict s) (f ⁻¹' t)
-- MeasureTheory.Measure.MeasurableSet.nullMeasurableSet_subtype_coe
theorem MeasurableSet.nullMeasurableSet_subtype_coe {t : Set s} (hs : NullMeasurableSet s μ)
    (ht : MeasurableSet t) : NullMeasurableSet ((↑) '' t) μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {t : Set ↑s} (hs : MeasureTheory.NullMeasurableSet s μ) (ht : MeasurableSet t)
Goal: MeasureTheory.NullMeasurableSet (Subtype.val '' t) μ
-- MeasureTheory.Measure.NullMeasurableSet.subtype_coe
theorem NullMeasurableSet.subtype_coe {t : Set s} (hs : NullMeasurableSet s μ)
    (ht : NullMeasurableSet t (μ.comap Subtype.val)) : NullMeasurableSet (((↑) : s → α) '' t) μ 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {t : Set ↑s} (hs : MeasureTheory.NullMeasurableSet s μ) (ht : MeasureTheory.NullMeasurableSet t (MeasureTheory.Measure.comap Subtype.val μ))
Goal: MeasureTheory.NullMeasurableSet (Subtype.val '' t) μ
-- MeasureTheory.Measure.measure_subtype_coe_le_comap
theorem measure_subtype_coe_le_comap (hs : NullMeasurableSet s μ) (t : Set s) :
    μ (((↑) : s → α) '' t) ≤ μ.comap Subtype.val t 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasureTheory.NullMeasurableSet s μ) (t : Set ↑s)
Goal: μ (Subtype.val '' t) ≤ (MeasureTheory.Measure.comap Subtype.val μ) t
-- MeasureTheory.Measure.measure_subtype_coe_eq_zero_of_comap_eq_zero
theorem measure_subtype_coe_eq_zero_of_comap_eq_zero (hs : NullMeasurableSet s μ) {t : Set s}
    (ht : μ.comap Subtype.val t = 0) : μ (((↑) : s → α) '' t) = 0 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasureTheory.NullMeasurableSet s μ) {t : Set ↑s} (ht : (MeasureTheory.Measure.comap Subtype.val μ) t = 0)
Goal: μ (Subtype.val '' t) = 0
-- MeasureTheory.Measure.Subtype.volume_def
theorem Subtype.volume_def : (volume : Measure u) = volume.comap Subtype.val 
Arguments: {δ : Type u_4} {u : Set δ} [MeasureTheory.MeasureSpace δ]
Goal: MeasureTheory.volume = MeasureTheory.Measure.comap Subtype.val MeasureTheory.volume
-- MeasureTheory.Measure.Subtype.volume_univ
theorem Subtype.volume_univ (hu : NullMeasurableSet u) : volume (univ : Set u) = volume u 
Arguments: {δ : Type u_4} {u : Set δ} [MeasureTheory.MeasureSpace δ] (hu : MeasureTheory.NullMeasurableSet u MeasureTheory.volume)
Goal: MeasureTheory.volume Set.univ = MeasureTheory.volume u
-- MeasureTheory.Measure.volume_subtype_coe_le_volume
theorem volume_subtype_coe_le_volume (hu : NullMeasurableSet u) (t : Set u) :
    volume (((↑) : u → δ) '' t) ≤ volume t 
Arguments: {δ : Type u_4} {u : Set δ} [MeasureTheory.MeasureSpace δ] (hu : MeasureTheory.NullMeasurableSet u MeasureTheory.volume) (t : Set ↑u)
Goal: MeasureTheory.volume (Subtype.val '' t) ≤ MeasureTheory.volume t
-- MeasureTheory.Measure.volume_subtype_coe_eq_zero_of_volume_eq_zero
theorem volume_subtype_coe_eq_zero_of_volume_eq_zero (hu : NullMeasurableSet u) {t : Set u}
    (ht : volume t = 0) : volume (((↑) : u → δ) '' t) = 0 
Arguments: {δ : Type u_4} {u : Set δ} [MeasureTheory.MeasureSpace δ] (hu : MeasureTheory.NullMeasurableSet u MeasureTheory.volume) {t : Set ↑u} (ht : MeasureTheory.volume t = 0)
Goal: MeasureTheory.volume (Subtype.val '' t) = 0
-- MeasurableEmbedding.map_comap
theorem map_comap (μ : Measure β) : (comap f μ).map f = μ.restrict (range f) 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} {m1 : MeasurableSpace β} {f : α → β} (hf : MeasurableEmbedding f) (μ : MeasureTheory.Measure β)
Goal: MeasureTheory.Measure.map f (MeasureTheory.Measure.comap f μ) = μ.restrict (Set.range f)
-- MeasurableEmbedding.comap_apply
theorem comap_apply (μ : Measure β) (s : Set α) : comap f μ s = μ (f '' s) 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} {m1 : MeasurableSpace β} {f : α → β} (hf : MeasurableEmbedding f) (μ : MeasureTheory.Measure β) (s : Set α)
Goal: (MeasureTheory.Measure.comap f μ) s = μ (f '' s)
-- MeasurableEmbedding.comap_map
theorem comap_map (μ : Measure α) : (map f μ).comap f = μ 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} {m1 : MeasurableSpace β} {f : α → β} (hf : MeasurableEmbedding f) (μ : MeasureTheory.Measure α)
Goal: MeasureTheory.Measure.comap f (MeasureTheory.Measure.map f μ) = μ
-- MeasurableEmbedding.ae_map_iff
theorem ae_map_iff {p : β → Prop} {μ : Measure α} : (∀ᵐ x ∂μ.map f, p x) ↔ ∀ᵐ x ∂μ, p (f x) 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} {m1 : MeasurableSpace β} {f : α → β} (hf : MeasurableEmbedding f) {p : β → Prop} {μ : MeasureTheory.Measure α}
Goal: (∀ᵐ (x : β) ∂MeasureTheory.Measure.map f μ, p x) ↔ ∀ᵐ (x : α) ∂μ, p (f x)
-- MeasurableEmbedding.restrict_map
theorem restrict_map (μ : Measure α) (s : Set β) :
    (μ.map f).restrict s = (μ.restrict <| f ⁻¹' s).map f 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} {m1 : MeasurableSpace β} {f : α → β} (hf : MeasurableEmbedding f) (μ : MeasureTheory.Measure α) (s : Set β)
Goal: (MeasureTheory.Measure.map f μ).restrict s = MeasureTheory.Measure.map f (μ.restrict (f ⁻¹' s))
-- MeasurableEmbedding.comap_preimage
protected theorem comap_preimage (μ : Measure β) (s : Set β) :
    μ.comap f (f ⁻¹' s) = μ (s ∩ range f) 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} {m1 : MeasurableSpace β} {f : α → β} (hf : MeasurableEmbedding f) (μ : MeasureTheory.Measure β) (s : Set β)
Goal: (MeasureTheory.Measure.comap f μ) (f ⁻¹' s) = μ (s ∩ Set.range f)
-- MeasurableEmbedding.comap_restrict
lemma comap_restrict (μ : Measure β) (s : Set β) :
    (μ.restrict s).comap f = (μ.comap f).restrict (f ⁻¹' s) 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} {m1 : MeasurableSpace β} {f : α → β} (hf : MeasurableEmbedding f) (μ : MeasureTheory.Measure β) (s : Set β)
Goal: MeasureTheory.Measure.comap f (μ.restrict s) = (MeasureTheory.Measure.comap f μ).restrict (f ⁻¹' s)
-- MeasurableEmbedding.restrict_comap
lemma restrict_comap (μ : Measure β) (s : Set α) :
    (μ.comap f).restrict s = (μ.restrict (f '' s)).comap f 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} {m1 : MeasurableSpace β} {f : α → β} (hf : MeasurableEmbedding f) (μ : MeasureTheory.Measure β) (s : Set α)
Goal: (MeasureTheory.Measure.comap f μ).restrict s = MeasureTheory.Measure.comap f (μ.restrict (f '' s))
-- MeasurableEquiv.restrict_map
theorem _root_.MeasurableEquiv.restrict_map (e : α ≃ᵐ β) (μ : Measure α) (s : Set β) :
    (μ.map e).restrict s = (μ.restrict <| e ⁻¹' s).map e 
Arguments: {α : Type u_2} {β : Type u_3} {m0 : MeasurableSpace α} {m1 : MeasurableSpace β} (e : α ≃ᵐ β) (μ : MeasureTheory.Measure α) (s : Set β)
Goal: (MeasureTheory.Measure.map (⇑e) μ).restrict s = MeasureTheory.Measure.map (⇑e) (μ.restrict (⇑e ⁻¹' s))
-- comap_subtype_coe_apply
theorem comap_subtype_coe_apply {_m0 : MeasurableSpace α} {s : Set α} (hs : MeasurableSet s)
    (μ : Measure α) (t : Set s) : comap (↑) μ t = μ ((↑) '' t) 
Arguments: {α : Type u_2} {_m0 : MeasurableSpace α} {s : Set α} (hs : MeasurableSet s) (μ : MeasureTheory.Measure α) (t : Set ↑s)
Goal: (MeasureTheory.Measure.comap Subtype.val μ) t = μ (Subtype.val '' t)
-- map_comap_subtype_coe
theorem map_comap_subtype_coe {m0 : MeasurableSpace α} {s : Set α} (hs : MeasurableSet s)
    (μ : Measure α) : (comap (↑) μ).map ((↑) : s → α) = μ.restrict s 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {s : Set α} (hs : MeasurableSet s) (μ : MeasureTheory.Measure α)
Goal: MeasureTheory.Measure.map Subtype.val (MeasureTheory.Measure.comap Subtype.val μ) = μ.restrict s
-- ae_restrict_iff_subtype
theorem ae_restrict_iff_subtype {m0 : MeasurableSpace α} {μ : Measure α} {s : Set α}
    (hs : MeasurableSet s) {p : α → Prop} :
    (∀ᵐ x ∂μ.restrict s, p x) ↔ ∀ᵐ (x : s) ∂comap ((↑) : s → α) μ, p x 
Arguments: {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) {p : α → Prop}
Goal: (∀ᵐ (x : α) ∂μ.restrict s, p x) ↔ ∀ᵐ (x : ↑s) ∂MeasureTheory.Measure.comap Subtype.val μ, p ↑x
-- volume_set_coe_def
theorem volume_set_coe_def (s : Set α) : (volume : Measure s) = comap ((↑) : s → α) volume 
Arguments: {α : Type u_2} [MeasureTheory.MeasureSpace α] (s : Set α)
Goal: MeasureTheory.volume = MeasureTheory.Measure.comap Subtype.val MeasureTheory.volume
-- MeasurableSet.map_coe_volume
theorem MeasurableSet.map_coe_volume {s : Set α} (hs : MeasurableSet s) :
    volume.map ((↑) : s → α) = restrict volume s 
Arguments: {α : Type u_2} [MeasureTheory.MeasureSpace α] {s : Set α} (hs : MeasurableSet s)
Goal: MeasureTheory.Measure.map Subtype.val MeasureTheory.volume = MeasureTheory.volume.restrict s
-- volume_image_subtype_coe
theorem volume_image_subtype_coe {s : Set α} (hs : MeasurableSet s) (t : Set s) :
    volume ((↑) '' t : Set α) = volume t 
Arguments: {α : Type u_2} [MeasureTheory.MeasureSpace α] {s : Set α} (hs : MeasurableSet s) (t : Set ↑s)
Goal: MeasureTheory.volume (Subtype.val '' t) = MeasureTheory.volume t
-- volume_preimage_coe
@[simp]
theorem volume_preimage_coe (hs : NullMeasurableSet s) (ht : MeasurableSet t) :
    volume (((↑) : s → α) ⁻¹' t) = volume (t ∩ s) 
Arguments: {α : Type u_2} [MeasureTheory.MeasureSpace α] {s t : Set α} (hs : MeasureTheory.NullMeasurableSet s MeasureTheory.volume) (ht : MeasurableSet t)
Goal: MeasureTheory.volume (Subtype.val ⁻¹' t) = MeasureTheory.volume (t ∩ s)
-- piecewise_ae_eq_restrict
theorem piecewise_ae_eq_restrict [DecidablePred (· ∈ s)] (hs : MeasurableSet s) :
    piecewise s f g =ᵐ[μ.restrict s] f 
Arguments: {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} {f g : α → β} [DecidablePred fun x => x ∈ s] (hs : MeasurableSet s)
Goal: s.piecewise f g =ᵐ[μ.restrict s] f
-- piecewise_ae_eq_restrict_compl
theorem piecewise_ae_eq_restrict_compl [DecidablePred (· ∈ s)] (hs : MeasurableSet s) :
    piecewise s f g =ᵐ[μ.restrict sᶜ] g 
Arguments: {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} {f g : α → β} [DecidablePred fun x => x ∈ s] (hs : MeasurableSet s)
Goal: s.piecewise f g =ᵐ[μ.restrict sᶜ] g
-- piecewise_ae_eq_of_ae_eq_set
theorem piecewise_ae_eq_of_ae_eq_set [DecidablePred (· ∈ s)] [DecidablePred (· ∈ t)]
    (hst : s =ᵐ[μ] t) : s.piecewise f g =ᵐ[μ] t.piecewise f g 
Arguments: {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s t : Set α} {f g : α → β} [DecidablePred fun x => x ∈ s] [DecidablePred fun x => x ∈ t] (hst : s =ᵐ[μ] t)
Goal: s.piecewise f g =ᵐ[μ] t.piecewise f g
-- mem_map_indicator_ae_iff_mem_map_restrict_ae_of_zero_mem
theorem mem_map_indicator_ae_iff_mem_map_restrict_ae_of_zero_mem [Zero β] {t : Set β}
    (ht : (0 : β) ∈ t) (hs : MeasurableSet s) :
    t ∈ Filter.map (s.indicator f) (ae μ) ↔ t ∈ Filter.map f (ae <| μ.restrict s) 
Arguments: {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} {f : α → β} [Zero β] {t : Set β} (ht : 0 ∈ t) (hs : MeasurableSet s)
Goal: t ∈ Filter.map (s.indicator f) (MeasureTheory.ae μ) ↔ t ∈ Filter.map f (MeasureTheory.ae (μ.restrict s))
-- mem_map_indicator_ae_iff_of_zero_nmem
theorem mem_map_indicator_ae_iff_of_zero_nmem [Zero β] {t : Set β} (ht : (0 : β) ∉ t) :
    t ∈ Filter.map (s.indicator f) (ae μ) ↔ μ ((f ⁻¹' t)ᶜ ∪ sᶜ) = 0 
Arguments: {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} {f : α → β} [Zero β] {t : Set β} (ht : 0 ∉ t)
Goal: t ∈ Filter.map (s.indicator f) (MeasureTheory.ae μ) ↔ μ ((f ⁻¹' t)ᶜ ∪ sᶜ) = 0
-- map_restrict_ae_le_map_indicator_ae
theorem map_restrict_ae_le_map_indicator_ae [Zero β] (hs : MeasurableSet s) :
    Filter.map f (ae <| μ.restrict s) ≤ Filter.map (s.indicator f) (ae μ) 
Arguments: {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} {f : α → β} [Zero β] (hs : MeasurableSet s)
Goal: Filter.map f (MeasureTheory.ae (μ.restrict s)) ≤ Filter.map (s.indicator f) (MeasureTheory.ae μ)
-- indicator_ae_eq_restrict
theorem indicator_ae_eq_restrict (hs : MeasurableSet s) : indicator s f =ᵐ[μ.restrict s] f 
Arguments: {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} {f : α → β} [Zero β] (hs : MeasurableSet s)
Goal: s.indicator f =ᵐ[μ.restrict s] f
-- indicator_ae_eq_restrict_compl
theorem indicator_ae_eq_restrict_compl (hs : MeasurableSet s) :
    indicator s f =ᵐ[μ.restrict sᶜ] 0 
Arguments: {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} {f : α → β} [Zero β] (hs : MeasurableSet s)
Goal: s.indicator f =ᵐ[μ.restrict sᶜ] 0
-- indicator_ae_eq_of_restrict_compl_ae_eq_zero
theorem indicator_ae_eq_of_restrict_compl_ae_eq_zero (hs : MeasurableSet s)
    (hf : f =ᵐ[μ.restrict sᶜ] 0) : s.indicator f =ᵐ[μ] f 
Arguments: {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} {f : α → β} [Zero β] (hs : MeasurableSet s) (hf : f =ᵐ[μ.restrict sᶜ] 0)
Goal: s.indicator f =ᵐ[μ] f
-- indicator_ae_eq_zero_of_restrict_ae_eq_zero
theorem indicator_ae_eq_zero_of_restrict_ae_eq_zero (hs : MeasurableSet s)
    (hf : f =ᵐ[μ.restrict s] 0) : s.indicator f =ᵐ[μ] 0 
Arguments: {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} {f : α → β} [Zero β] (hs : MeasurableSet s) (hf : f =ᵐ[μ.restrict s] 0)
Goal: s.indicator f =ᵐ[μ] 0
-- indicator_ae_eq_of_ae_eq_set
theorem indicator_ae_eq_of_ae_eq_set (hst : s =ᵐ[μ] t) : s.indicator f =ᵐ[μ] t.indicator f 
Arguments: {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s t : Set α} {f : α → β} [Zero β] (hst : s =ᵐ[μ] t)
Goal: s.indicator f =ᵐ[μ] t.indicator f
-- indicator_meas_zero
theorem indicator_meas_zero (hs : μ s = 0) : indicator s f =ᵐ[μ] 0 
Arguments: {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} {f : α → β} [Zero β] (hs : μ s = 0)
Goal: s.indicator f =ᵐ[μ] 0
-- ae_eq_restrict_iff_indicator_ae_eq
theorem ae_eq_restrict_iff_indicator_ae_eq {g : α → β} (hs : MeasurableSet s) :
    f =ᵐ[μ.restrict s] g ↔ s.indicator f =ᵐ[μ] s.indicator g 
Arguments: {α : Type u_2} {β : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} {f : α → β} [Zero β] {g : α → β} (hs : MeasurableSet s)
Goal: f =ᵐ[μ.restrict s] g ↔ s.indicator f =ᵐ[μ] s.indicator g
-- MeasureTheory.Measure.IsAddLeftInvariant.map_add_left_eq_self
class IsAddLeftInvariant [Add G] (μ : Measure G) : Prop where
  map_add_left_eq_self : ∀ g : G, map (g + ·) μ = μ
Arguments: {G : Type u_2} [MeasurableSpace G] [Add G] {μ : MeasureTheory.Measure G} [self : μ.IsAddLeftInvariant] (g : G)
Goal: MeasureTheory.Measure.map (fun x => g + x) μ = μ
-- MeasureTheory.Measure.IsMulLeftInvariant.map_mul_left_eq_self
@[to_additive existing]
class IsMulLeftInvariant [Mul G] (μ : Measure G) : Prop where
  map_mul_left_eq_self : ∀ g : G, map (g * ·) μ = μ
Arguments: {G : Type u_2} [MeasurableSpace G] [Mul G] {μ : MeasureTheory.Measure G} [self : μ.IsMulLeftInvariant] (g : G)
Goal: MeasureTheory.Measure.map (fun x => g * x) μ = μ