ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Measure.map_toOuterMeasure
theorem map_toOuterMeasure (hf : AEMeasurable f μ) :
    (μ.map f).toOuterMeasure = (OuterMeasure.map f μ.toOuterMeasure).trim 
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.map f μ).toOuterMeasure = ((MeasureTheory.OuterMeasure.map f) μ.toOuterMeasure).trim
-- MeasureTheory.Measure.map_eq_zero_iff
@[simp] lemma map_eq_zero_iff (hf : AEMeasurable f μ) : μ.map f = 0 ↔ μ = 0 
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.map f μ = 0 ↔ μ = 0
-- MeasureTheory.Measure.mapₗ_eq_zero_iff
@[simp] lemma mapₗ_eq_zero_iff (hf : Measurable f) : Measure.mapₗ f μ = 0 ↔ μ = 0 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β} (hf : Measurable f)
Goal: (MeasureTheory.Measure.mapₗ f) μ = 0 ↔ μ = 0
-- MeasureTheory.Measure.measure_preimage_of_map_eq_self
lemma measure_preimage_of_map_eq_self {f : α → α} (hf : map f μ = μ)
    {s : Set α} (hs : NullMeasurableSet s μ) : μ (f ⁻¹' s) = μ s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → α} (hf : MeasureTheory.Measure.map f μ = μ) {s : Set α} (hs : MeasureTheory.NullMeasurableSet s μ)
Goal: μ (f ⁻¹' s) = μ s
-- MeasureTheory.Measure.map_ne_zero_iff
lemma map_ne_zero_iff (hf : AEMeasurable f μ) : μ.map f ≠ 0 ↔ μ ≠ 0 
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.map f μ ≠ 0 ↔ μ ≠ 0
-- MeasureTheory.Measure.mapₗ_ne_zero_iff
lemma mapₗ_ne_zero_iff (hf : Measurable f) : Measure.mapₗ f μ ≠ 0 ↔ μ ≠ 0 
Arguments: {α : Type u_1} {β : Type u_2} {m0 : MeasurableSpace α} [MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β} (hf : Measurable f)
Goal: (MeasureTheory.Measure.mapₗ f) μ ≠ 0 ↔ μ ≠ 0
-- MeasureTheory.Measure.map_id
@[simp]
theorem map_id : map id μ = μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: MeasureTheory.Measure.map id μ = μ
-- MeasureTheory.Measure.map_id'
@[simp]
theorem map_id' : map (fun x => x) μ = μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: MeasureTheory.Measure.map (fun x => x) μ = μ
-- MeasureTheory.Measure.map_map
theorem map_map {g : β → γ} {f : α → β} (hg : Measurable g) (hf : Measurable f) :
    (μ.map f).map g = μ.map (g ∘ f) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {g : β → γ} {f : α → β} (hg : Measurable g) (hf : Measurable f)
Goal: MeasureTheory.Measure.map g (MeasureTheory.Measure.map f μ) = MeasureTheory.Measure.map (g ∘ f) μ
-- MeasureTheory.Measure.map_mono
@[mono]
theorem map_mono {f : α → β} (h : μ ≤ ν) (hf : Measurable 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} (h : μ ≤ ν) (hf : Measurable f)
Goal: MeasureTheory.Measure.map f μ ≤ MeasureTheory.Measure.map f ν
-- MeasureTheory.Measure.le_map_apply
theorem le_map_apply {f : α → β} (hf : AEMeasurable f μ) (s : Set β) : μ (f ⁻¹' s) ≤ μ.map f 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)
Goal: μ (f ⁻¹' s) ≤ (MeasureTheory.Measure.map f μ) s
-- MeasureTheory.Measure.le_map_apply_image
theorem le_map_apply_image {f : α → β} (hf : AEMeasurable f μ) (s : Set α) :
    μ s ≤ μ.map f (f '' 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 α)
Goal: μ s ≤ (MeasureTheory.Measure.map f μ) (f '' s)
-- MeasureTheory.Measure.preimage_null_of_map_null
theorem preimage_null_of_map_null {f : α → β} (hf : AEMeasurable f μ) {s : Set β}
    (hs : μ.map f s = 0) : μ (f ⁻¹' s) = 0 
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.Measure.map f μ) s = 0)
Goal: μ (f ⁻¹' s) = 0
-- MeasureTheory.Measure.tendsto_ae_map
theorem tendsto_ae_map {f : α → β} (hf : AEMeasurable f μ) : Tendsto f (ae μ) (ae (μ.map f)) 
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: Filter.Tendsto f (MeasureTheory.ae μ) (MeasureTheory.ae (MeasureTheory.Measure.map f μ))
-- MeasureTheory.Measure.comapₗ_apply
theorem comapₗ_apply {β} [MeasurableSpace α] {mβ : MeasurableSpace β} (f : α → β)
    (hfi : Injective f) (hf : ∀ s, MeasurableSet s → MeasurableSet (f '' s)) (μ : Measure β)
    (hs : MeasurableSet s) : comapₗ f μ s = μ (f '' s) 
Arguments: {α : Type u_1} {s : Set α} {β : Type u_8} [MeasurableSpace α] {mβ : MeasurableSpace β} (f : α → β) (hfi : Function.Injective f) (hf : ∀ (s : Set α), MeasurableSet s → MeasurableSet (f '' s)) (μ : MeasureTheory.Measure β) (hs : MeasurableSet s)
Goal: ((MeasureTheory.Measure.comapₗ f) μ) s = μ (f '' s)
-- MeasureTheory.Measure.comap_apply₀
theorem comap_apply₀ [MeasurableSpace α] (f : α → β) (μ : Measure β) (hfi : Injective f)
    (hf : ∀ s, MeasurableSet s → NullMeasurableSet (f '' s) μ)
    (hs : NullMeasurableSet s (comap f μ)) : comap f μ s = μ (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace β] {s : Set α} [MeasurableSpace α] (f : α → β) (μ : MeasureTheory.Measure β) (hfi : Function.Injective f) (hf : ∀ (s : Set α), MeasurableSet s → MeasureTheory.NullMeasurableSet (f '' s) μ) (hs : MeasureTheory.NullMeasurableSet s (MeasureTheory.Measure.comap f μ))
Goal: (MeasureTheory.Measure.comap f μ) s = μ (f '' s)
-- MeasureTheory.Measure.le_comap_apply
theorem le_comap_apply {β} [MeasurableSpace α] {mβ : MeasurableSpace β} (f : α → β) (μ : Measure β)
    (hfi : Injective f) (hf : ∀ s, MeasurableSet s → NullMeasurableSet (f '' s) μ) (s : Set α) :
    μ (f '' s) ≤ comap f μ s 
Arguments: {α : Type u_1} {β : Type u_8} [MeasurableSpace α] {mβ : MeasurableSpace β} (f : α → β) (μ : MeasureTheory.Measure β) (hfi : Function.Injective f) (hf : ∀ (s : Set α), MeasurableSet s → MeasureTheory.NullMeasurableSet (f '' s) μ) (s : Set α)
Goal: μ (f '' s) ≤ (MeasureTheory.Measure.comap f μ) s
-- MeasureTheory.Measure.comap_apply
theorem comap_apply {β} [MeasurableSpace α] {_mβ : MeasurableSpace β} (f : α → β)
    (hfi : Injective f) (hf : ∀ s, MeasurableSet s → MeasurableSet (f '' s)) (μ : Measure β)
    (hs : MeasurableSet s) : comap f μ s = μ (f '' s) 
Arguments: {α : Type u_1} {s : Set α} {β : Type u_8} [MeasurableSpace α] {_mβ : MeasurableSpace β} (f : α → β) (hfi : Function.Injective f) (hf : ∀ (s : Set α), MeasurableSet s → MeasurableSet (f '' s)) (μ : MeasureTheory.Measure β) (hs : MeasurableSet s)
Goal: (MeasureTheory.Measure.comap f μ) s = μ (f '' s)
-- MeasureTheory.Measure.comapₗ_eq_comap
theorem comapₗ_eq_comap {β} [MeasurableSpace α] {_mβ : MeasurableSpace β} (f : α → β)
    (hfi : Injective f) (hf : ∀ s, MeasurableSet s → MeasurableSet (f '' s)) (μ : Measure β)
    (hs : MeasurableSet s) : comapₗ f μ s = comap f μ s 
Arguments: {α : Type u_1} {s : Set α} {β : Type u_8} [MeasurableSpace α] {_mβ : MeasurableSpace β} (f : α → β) (hfi : Function.Injective f) (hf : ∀ (s : Set α), MeasurableSet s → MeasurableSet (f '' s)) (μ : MeasureTheory.Measure β) (hs : MeasurableSet s)
Goal: ((MeasureTheory.Measure.comapₗ f) μ) s = (MeasureTheory.Measure.comap f μ) s
-- MeasureTheory.Measure.measure_image_eq_zero_of_comap_eq_zero
theorem measure_image_eq_zero_of_comap_eq_zero {β} [MeasurableSpace α] {_mβ : MeasurableSpace β}
    (f : α → β) (μ : Measure β) (hfi : Injective f)
    (hf : ∀ s, MeasurableSet s → NullMeasurableSet (f '' s) μ) {s : Set α} (hs : comap f μ s = 0) :
    μ (f '' s) = 0 
Arguments: {α : Type u_1} {β : Type u_8} [MeasurableSpace α] {_mβ : MeasurableSpace β} (f : α → β) (μ : MeasureTheory.Measure β) (hfi : Function.Injective f) (hf : ∀ (s : Set α), MeasurableSet s → MeasureTheory.NullMeasurableSet (f '' s) μ) {s : Set α} (hs : (MeasureTheory.Measure.comap f μ) s = 0)
Goal: μ (f '' s) = 0
-- MeasureTheory.Measure.ae_eq_image_of_ae_eq_comap
theorem ae_eq_image_of_ae_eq_comap {β} [MeasurableSpace α] {mβ : MeasurableSpace β} (f : α → β)
    (μ : Measure β) (hfi : Injective f) (hf : ∀ s, MeasurableSet s → NullMeasurableSet (f '' s) μ)
    {s t : Set α} (hst : s =ᵐ[comap f μ] t) : f '' s =ᵐ[μ] f '' t 
Arguments: {α : Type u_1} {β : Type u_8} [MeasurableSpace α] {mβ : MeasurableSpace β} (f : α → β) (μ : MeasureTheory.Measure β) (hfi : Function.Injective f) (hf : ∀ (s : Set α), MeasurableSet s → MeasureTheory.NullMeasurableSet (f '' s) μ) {s t : Set α} (hst : s =ᵐ[MeasureTheory.Measure.comap f μ] t)
Goal: f '' s =ᵐ[μ] f '' t
-- MeasureTheory.Measure.NullMeasurableSet.image
theorem NullMeasurableSet.image {β} [MeasurableSpace α] {mβ : MeasurableSpace β} (f : α → β)
    (μ : Measure β) (hfi : Injective f) (hf : ∀ s, MeasurableSet s → NullMeasurableSet (f '' s) μ)
    {s : Set α} (hs : NullMeasurableSet s (μ.comap f)) : NullMeasurableSet (f '' s) μ 
Arguments: {α : Type u_1} {β : Type u_8} [MeasurableSpace α] {mβ : MeasurableSpace β} (f : α → β) (μ : MeasureTheory.Measure β) (hfi : Function.Injective f) (hf : ∀ (s : Set α), MeasurableSet s → MeasureTheory.NullMeasurableSet (f '' s) μ) {s : Set α} (hs : MeasureTheory.NullMeasurableSet s (MeasureTheory.Measure.comap f μ))
Goal: MeasureTheory.NullMeasurableSet (f '' s) μ
-- MeasureTheory.Measure.comap_preimage
theorem comap_preimage {β} [MeasurableSpace α] {mβ : MeasurableSpace β} (f : α → β) (μ : Measure β)
    {s : Set β} (hf : Injective f) (hf' : Measurable f)
    (h : ∀ t, MeasurableSet t → NullMeasurableSet (f '' t) μ) (hs : MeasurableSet s) :
    μ.comap f (f ⁻¹' s) = μ (s ∩ range f) 
Arguments: {α : Type u_1} {β : Type u_8} [MeasurableSpace α] {mβ : MeasurableSpace β} (f : α → β) (μ : MeasureTheory.Measure β) {s : Set β} (hf : Function.Injective f) (hf' : Measurable f) (h : ∀ (t : Set α), MeasurableSet t → MeasureTheory.NullMeasurableSet (f '' t) μ) (hs : MeasurableSet s)
Goal: (MeasureTheory.Measure.comap f μ) (f ⁻¹' s) = μ (s ∩ Set.range f)
-- MeasureTheory.Measure.le_sum_apply
theorem le_sum_apply (f : ι → Measure α) (s : Set α) : ∑' i, f i s ≤ sum f s 
Arguments: {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} (f : ι → MeasureTheory.Measure α) (s : Set α)
Goal: ∑' (i : ι), (f i) s ≤ (MeasureTheory.Measure.sum f) s
-- MeasureTheory.Measure.sum_apply
@[simp]
theorem sum_apply (f : ι → Measure α) {s : Set α} (hs : MeasurableSet s) :
    sum f s = ∑' i, f i s 
Arguments: {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} (f : ι → MeasureTheory.Measure α) {s : Set α} (hs : MeasurableSet s)
Goal: (MeasureTheory.Measure.sum f) s = ∑' (i : ι), (f i) s
-- MeasureTheory.Measure.sum_apply₀
theorem sum_apply₀ (f : ι → Measure α) {s : Set α} (hs : NullMeasurableSet s (sum f)) :
    sum f s = ∑' i, f i s 
Arguments: {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} (f : ι → MeasureTheory.Measure α) {s : Set α} (hs : MeasureTheory.NullMeasurableSet s (MeasureTheory.Measure.sum f))
Goal: (MeasureTheory.Measure.sum f) s = ∑' (i : ι), (f i) s
-- MeasureTheory.Measure.sum_apply_of_countable
theorem sum_apply_of_countable [Countable ι] (f : ι → Measure α) (s : Set α) :
    sum f s = ∑' i, f i s 
Arguments: {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} [Countable ι] (f : ι → MeasureTheory.Measure α) (s : Set α)
Goal: (MeasureTheory.Measure.sum f) s = ∑' (i : ι), (f i) s
-- MeasureTheory.Measure.le_sum
theorem le_sum (μ : ι → Measure α) (i : ι) : μ i ≤ sum μ 
Arguments: {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} (μ : ι → MeasureTheory.Measure α) (i : ι)
Goal: μ i ≤ MeasureTheory.Measure.sum μ
-- MeasureTheory.Measure.sum_apply_eq_zero
@[simp]
theorem sum_apply_eq_zero [Countable ι] {μ : ι → Measure α} {s : Set α} :
    sum μ s = 0 ↔ ∀ i, μ i s = 0 
Arguments: {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} [Countable ι] {μ : ι → MeasureTheory.Measure α} {s : Set α}
Goal: (MeasureTheory.Measure.sum μ) s = 0 ↔ ∀ (i : ι), (μ i) s = 0
-- MeasureTheory.Measure.sum_apply_eq_zero'
theorem sum_apply_eq_zero' {μ : ι → Measure α} {s : Set α} (hs : MeasurableSet s) :
    sum μ s = 0 ↔ ∀ i, μ i s = 0 
Arguments: {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} {μ : ι → MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s)
Goal: (MeasureTheory.Measure.sum μ) s = 0 ↔ ∀ (i : ι), (μ i) s = 0
-- MeasureTheory.Measure.sum_zero
@[simp]
lemma sum_zero : Measure.sum (fun (_ : ι) ↦ (0 : Measure α)) = 0 
Arguments: {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α}
Goal: (MeasureTheory.Measure.sum fun x => 0) = 0
-- MeasureTheory.Measure.sum_sum
theorem sum_sum {ι' : Type*} (μ : ι → ι' → Measure α) :
    (sum fun n => sum (μ n)) = sum (fun (p : ι × ι') ↦ μ p.1 p.2) 
Arguments: {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} {ι' : Type u_8} (μ : ι → ι' → MeasureTheory.Measure α)
Goal: (MeasureTheory.Measure.sum fun n => MeasureTheory.Measure.sum (μ n)) = MeasureTheory.Measure.sum fun p => μ p.1 p.2
-- MeasureTheory.Measure.sum_comm
theorem sum_comm {ι' : Type*} (μ : ι → ι' → Measure α) :
    (sum fun n => sum (μ n)) = sum fun m => sum fun n => μ n m 
Arguments: {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} {ι' : Type u_8} (μ : ι → ι' → MeasureTheory.Measure α)
Goal: (MeasureTheory.Measure.sum fun n => MeasureTheory.Measure.sum (μ n)) =
  MeasureTheory.Measure.sum fun m => MeasureTheory.Measure.sum fun n => μ n m
-- MeasureTheory.Measure.ae_sum_iff
theorem ae_sum_iff [Countable ι] {μ : ι → Measure α} {p : α → Prop} :
    (∀ᵐ x ∂sum μ, p x) ↔ ∀ i, ∀ᵐ x ∂μ i, p x 
Arguments: {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} [Countable ι] {μ : ι → MeasureTheory.Measure α} {p : α → Prop}
Goal: (∀ᵐ (x : α) ∂MeasureTheory.Measure.sum μ, p x) ↔ ∀ (i : ι), ∀ᵐ (x : α) ∂μ i, p x
-- MeasureTheory.Measure.ae_sum_iff'
theorem ae_sum_iff' {μ : ι → Measure α} {p : α → Prop} (h : MeasurableSet { x | p x }) :
    (∀ᵐ x ∂sum μ, p x) ↔ ∀ i, ∀ᵐ x ∂μ i, p x 
Arguments: {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} {μ : ι → MeasureTheory.Measure α} {p : α → Prop} (h : MeasurableSet {x | p x})
Goal: (∀ᵐ (x : α) ∂MeasureTheory.Measure.sum μ, p x) ↔ ∀ (i : ι), ∀ᵐ (x : α) ∂μ i, p x
-- MeasureTheory.Measure.sum_fintype
@[simp]
theorem sum_fintype [Fintype ι] (μ : ι → Measure α) : sum μ = ∑ i, μ i 
Arguments: {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} [Fintype ι] (μ : ι → MeasureTheory.Measure α)
Goal: MeasureTheory.Measure.sum μ = ∑ i : ι, μ i
-- MeasureTheory.Measure.sum_coe_finset
theorem sum_coe_finset (s : Finset ι) (μ : ι → Measure α) :
    (sum fun i : s => μ i) = ∑ i ∈ s, μ i 
Arguments: {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} (s : Finset ι) (μ : ι → MeasureTheory.Measure α)
Goal: (MeasureTheory.Measure.sum fun i => μ ↑i) = ∑ i ∈ s, μ i
-- MeasureTheory.Measure.ae_sum_eq
@[simp]
theorem ae_sum_eq [Countable ι] (μ : ι → Measure α) : ae (sum μ) = ⨆ i, ae (μ i) 
Arguments: {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} [Countable ι] (μ : ι → MeasureTheory.Measure α)
Goal: MeasureTheory.ae (MeasureTheory.Measure.sum μ) = ⨆ i, MeasureTheory.ae (μ i)
-- MeasureTheory.Measure.sum_bool
theorem sum_bool (f : Bool → Measure α) : sum f = f true + f false 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (f : Bool → MeasureTheory.Measure α)
Goal: MeasureTheory.Measure.sum f = f true + f false
-- MeasureTheory.Measure.sum_cond
theorem sum_cond (μ ν : Measure α) : (sum fun b => cond b μ ν) = μ + ν 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ ν : MeasureTheory.Measure α)
Goal: (MeasureTheory.Measure.sum fun b => bif b then μ else ν) = μ + ν
-- MeasureTheory.Measure.sum_of_isEmpty
@[simp]
theorem sum_of_isEmpty [IsEmpty ι] (μ : ι → Measure α) : sum μ = 0 
Arguments: {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} [IsEmpty ι] (μ : ι → MeasureTheory.Measure α)
Goal: MeasureTheory.Measure.sum μ = 0
-- MeasureTheory.Measure.sum_add_sum_compl
theorem sum_add_sum_compl (s : Set ι) (μ : ι → Measure α) :
    ((sum fun i : s => μ i) + sum fun i : ↥sᶜ => μ i) = sum μ 
Arguments: {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} (s : Set ι) (μ : ι → MeasureTheory.Measure α)
Goal: ((MeasureTheory.Measure.sum fun i => μ ↑i) + MeasureTheory.Measure.sum fun i => μ ↑i) = MeasureTheory.Measure.sum μ
-- MeasureTheory.Measure.sum_congr
theorem sum_congr {μ ν : ℕ → Measure α} (h : ∀ n, μ n = ν n) : sum μ = sum ν 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : ℕ → MeasureTheory.Measure α} (h : ∀ (n : ℕ), μ n = ν n)
Goal: MeasureTheory.Measure.sum μ = MeasureTheory.Measure.sum ν
-- MeasureTheory.Measure.sum_add_sum
theorem sum_add_sum {ι : Type*} (μ ν : ι → Measure α) : sum μ + sum ν = sum fun n => μ n + ν n 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {ι : Type u_8} (μ ν : ι → MeasureTheory.Measure α)
Goal: MeasureTheory.Measure.sum μ + MeasureTheory.Measure.sum ν = MeasureTheory.Measure.sum fun n => μ n + ν n
-- MeasureTheory.Measure.sum_comp_equiv
@[simp] lemma sum_comp_equiv {ι ι' : Type*} (e : ι' ≃ ι) (m : ι → Measure α) :
    sum (m ∘ e) = sum m 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {ι : Type u_8} {ι' : Type u_9} (e : ι' ≃ ι) (m : ι → MeasureTheory.Measure α)
Goal: MeasureTheory.Measure.sum (m ∘ ⇑e) = MeasureTheory.Measure.sum m
-- MeasureTheory.Measure.sum_extend_zero
@[simp] lemma sum_extend_zero {ι ι' : Type*} {f : ι → ι'} (hf : Injective f) (m : ι → Measure α) :
    sum (Function.extend f m 0) = sum m 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {ι : Type u_8} {ι' : Type u_9} {f : ι → ι'} (hf : Function.Injective f) (m : ι → MeasureTheory.Measure α)
Goal: MeasureTheory.Measure.sum (Function.extend f m 0) = MeasureTheory.Measure.sum m
-- MeasureTheory.Measure.absolutelyContinuous_of_le
theorem absolutelyContinuous_of_le (h : μ ≤ ν) : μ ≪ ν 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} (h : μ ≤ ν)
Goal: μ.AbsolutelyContinuous ν
-- MeasureTheory.Measure.absolutelyContinuous_of_eq
theorem absolutelyContinuous_of_eq (h : μ = ν) : μ ≪ ν 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} (h : μ = ν)
Goal: μ.AbsolutelyContinuous ν
-- MeasureTheory.Measure.AbsolutelyContinuous.mk
theorem mk (h : ∀ ⦃s : Set α⦄, MeasurableSet s → ν s = 0 → μ s = 0) : μ ≪ ν 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} (h : ∀ ⦃s : Set α⦄, MeasurableSet s → ν s = 0 → μ s = 0)
Goal: μ.AbsolutelyContinuous ν
-- MeasureTheory.Measure.AbsolutelyContinuous.refl
@[refl]
protected theorem refl {_m0 : MeasurableSpace α} (μ : Measure α) : μ ≪ μ 
Arguments: {α : Type u_1} {_m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α)
Goal: μ.AbsolutelyContinuous μ
-- MeasureTheory.Measure.AbsolutelyContinuous.rfl
protected theorem rfl : μ ≪ μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: μ.AbsolutelyContinuous μ
-- MeasureTheory.Measure.AbsolutelyContinuous.zero
@[simp]
protected lemma zero (μ : Measure α) : 0 ≪ μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α)
Goal: MeasureTheory.Measure.AbsolutelyContinuous 0 μ
-- MeasureTheory.Measure.AbsolutelyContinuous.trans
@[trans]
protected theorem trans (h1 : μ₁ ≪ μ₂) (h2 : μ₂ ≪ μ₃) : μ₁ ≪ μ₃ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ₁ μ₂ μ₃ : MeasureTheory.Measure α} (h1 : μ₁.AbsolutelyContinuous μ₂) (h2 : μ₂.AbsolutelyContinuous μ₃)
Goal: μ₁.AbsolutelyContinuous μ₃
-- MeasureTheory.Measure.AbsolutelyContinuous.map
@[mono]
protected theorem map (h : μ ≪ ν) {f : α → β} (hf : Measurable 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} (h : μ.AbsolutelyContinuous ν) {f : α → β} (hf : Measurable f)
Goal: (MeasureTheory.Measure.map f μ).AbsolutelyContinuous (MeasureTheory.Measure.map f ν)
-- MeasureTheory.Measure.AbsolutelyContinuous.smul
protected theorem smul [Monoid R] [DistribMulAction R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞]
    (h : μ ≪ ν) (c : R) : c • μ ≪ ν 
Arguments: {α : Type u_1} {R : Type u_6} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [Monoid R] [DistribMulAction R ENNReal] [IsScalarTower R ENNReal ENNReal] (h : μ.AbsolutelyContinuous ν) (c : R)
Goal: (c • μ).AbsolutelyContinuous ν
-- MeasureTheory.Measure.AbsolutelyContinuous.add
protected lemma add (h1 : μ₁ ≪ ν) (h2 : μ₂ ≪ ν') : μ₁ + μ₂ ≪ ν + ν' 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ₁ μ₂ ν ν' : MeasureTheory.Measure α} (h1 : μ₁.AbsolutelyContinuous ν) (h2 : μ₂.AbsolutelyContinuous ν')
Goal: (μ₁ + μ₂).AbsolutelyContinuous (ν + ν')
-- MeasureTheory.Measure.AbsolutelyContinuous.add_left_iff
lemma add_left_iff {μ₁ μ₂ ν : Measure α} :
    μ₁ + μ₂ ≪ ν ↔ μ₁ ≪ ν ∧ μ₂ ≪ ν 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ₁ μ₂ ν : MeasureTheory.Measure α}
Goal: (μ₁ + μ₂).AbsolutelyContinuous ν ↔ μ₁.AbsolutelyContinuous ν ∧ μ₂.AbsolutelyContinuous ν
-- MeasureTheory.Measure.AbsolutelyContinuous.add_left
lemma add_left {μ₁ μ₂ ν : Measure α} (h₁ : μ₁ ≪ ν) (h₂ : μ₂ ≪ ν) : μ₁ + μ₂ ≪ ν 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ₁ μ₂ ν : MeasureTheory.Measure α} (h₁ : μ₁.AbsolutelyContinuous ν) (h₂ : μ₂.AbsolutelyContinuous ν)
Goal: (μ₁ + μ₂).AbsolutelyContinuous ν
-- MeasureTheory.Measure.AbsolutelyContinuous.add_right
lemma add_right (h1 : μ ≪ ν) (ν' : Measure α) : μ ≪ ν + ν' 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} (h1 : μ.AbsolutelyContinuous ν) (ν' : MeasureTheory.Measure α)
Goal: μ.AbsolutelyContinuous (ν + ν')
-- MeasureTheory.Measure.absolutelyContinuous_zero_iff
@[simp]
lemma absolutelyContinuous_zero_iff : μ ≪ 0 ↔ μ = 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: μ.AbsolutelyContinuous 0 ↔ μ = 0
-- MeasureTheory.Measure.absolutelyContinuous_sum_left
lemma absolutelyContinuous_sum_left {μs : ι → Measure α} (hμs : ∀ i, μs i ≪ ν) :
    Measure.sum μs ≪ ν 
Arguments: {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} {ν : MeasureTheory.Measure α} {μs : ι → MeasureTheory.Measure α} (hμs : ∀ (i : ι), (μs i).AbsolutelyContinuous ν)
Goal: (MeasureTheory.Measure.sum μs).AbsolutelyContinuous ν
-- MeasureTheory.Measure.absolutelyContinuous_sum_right
lemma absolutelyContinuous_sum_right {μs : ι → Measure α} (i : ι) (hνμ : ν ≪ μs i) :
    ν ≪ Measure.sum μs 
Arguments: {α : Type u_1} {ι : Type u_5} {m0 : MeasurableSpace α} {ν : MeasureTheory.Measure α} {μs : ι → MeasureTheory.Measure α} (i : ι) (hνμ : ν.AbsolutelyContinuous (μs i))
Goal: ν.AbsolutelyContinuous (MeasureTheory.Measure.sum μs)
-- MeasureTheory.Measure.absolutelyContinuous_of_le_smul
theorem absolutelyContinuous_of_le_smul {μ' : Measure α} {c : ℝ≥0∞} (hμ'_le : μ' ≤ c • μ) :
    μ' ≪ μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ μ' : MeasureTheory.Measure α} {c : ENNReal} (hμ'_le : μ' ≤ c • μ)
Goal: μ'.AbsolutelyContinuous μ
-- MeasureTheory.Measure.smul_absolutelyContinuous
lemma smul_absolutelyContinuous {c : ℝ≥0∞} : c • μ ≪ μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {c : ENNReal}
Goal: (c • μ).AbsolutelyContinuous μ
-- MeasureTheory.Measure.absolutelyContinuous_smul
lemma absolutelyContinuous_smul {c : ℝ≥0∞} (hc : c ≠ 0) : μ ≪ c • μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {c : ENNReal} (hc : c ≠ 0)
Goal: μ.AbsolutelyContinuous (c • μ)
-- MeasureTheory.Measure.ae_le_iff_absolutelyContinuous
theorem ae_le_iff_absolutelyContinuous : ae μ ≤ ae ν ↔ μ ≪ ν 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α}
Goal: MeasureTheory.ae μ ≤ MeasureTheory.ae ν ↔ μ.AbsolutelyContinuous ν