ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Measurable.min
@[measurability]
theorem Measurable.min {f g : δ → α} (hf : Measurable f) (hg : Measurable g) :
    Measurable fun a => min (f a) (g a) 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderClosedTopology α] [SecondCountableTopology α] {f g : δ → α} (hf : Measurable f) (hg : Measurable g)
Goal: Measurable fun a => min (f a) (g a)
-- AEMeasurable.min
@[measurability]
nonrec theorem AEMeasurable.min {f g : δ → α} {μ : Measure δ} (hf : AEMeasurable f μ)
    (hg : AEMeasurable g μ) : AEMeasurable (fun a => min (f a) (g a)) μ 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderClosedTopology α] [SecondCountableTopology α] {f g : δ → α} {μ : MeasureTheory.Measure δ} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: AEMeasurable (fun a => min (f a) (g a)) μ
-- measurable_of_Iio
theorem measurable_of_Iio {f : δ → α} (hf : ∀ x, MeasurableSet (f ⁻¹' Iio x)) : Measurable f 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] {f : δ → α} (hf : ∀ (x : α), MeasurableSet (f ⁻¹' Set.Iio x))
Goal: Measurable f
-- UpperSemicontinuous.measurable
theorem UpperSemicontinuous.measurable [TopologicalSpace δ] [OpensMeasurableSpace δ] {f : δ → α}
    (hf : UpperSemicontinuous f) : Measurable f 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] [TopologicalSpace δ] [OpensMeasurableSpace δ] {f : δ → α} (hf : UpperSemicontinuous f)
Goal: Measurable f
-- measurable_of_Ioi
theorem measurable_of_Ioi {f : δ → α} (hf : ∀ x, MeasurableSet (f ⁻¹' Ioi x)) : Measurable f 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] {f : δ → α} (hf : ∀ (x : α), MeasurableSet (f ⁻¹' Set.Ioi x))
Goal: Measurable f
-- LowerSemicontinuous.measurable
theorem LowerSemicontinuous.measurable [TopologicalSpace δ] [OpensMeasurableSpace δ] {f : δ → α}
    (hf : LowerSemicontinuous f) : Measurable f 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] [TopologicalSpace δ] [OpensMeasurableSpace δ] {f : δ → α} (hf : LowerSemicontinuous f)
Goal: Measurable f
-- measurable_of_Iic
theorem measurable_of_Iic {f : δ → α} (hf : ∀ x, MeasurableSet (f ⁻¹' Iic x)) : Measurable f 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] {f : δ → α} (hf : ∀ (x : α), MeasurableSet (f ⁻¹' Set.Iic x))
Goal: Measurable f
-- measurable_of_Ici
theorem measurable_of_Ici {f : δ → α} (hf : ∀ x, MeasurableSet (f ⁻¹' Ici x)) : Measurable f 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] {f : δ → α} (hf : ∀ (x : α), MeasurableSet (f ⁻¹' Set.Ici x))
Goal: Measurable f
-- Measurable.isLUB
theorem Measurable.isLUB {ι} [Countable ι] {f : ι → δ → α} {g : δ → α} (hf : ∀ i, Measurable (f i))
    (hg : ∀ b, IsLUB { a | ∃ i, f i b = a } (g b)) : Measurable g 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Sort u_5} [Countable ι] {f : ι → δ → α} {g : δ → α} (hf : ∀ (i : ι), Measurable (f i)) (hg : ∀ (b : δ), IsLUB {a | ∃ i, f i b = a} (g b))
Goal: Measurable g
-- Measurable.isLUB_of_mem
theorem Measurable.isLUB_of_mem {ι} [Countable ι] {f : ι → δ → α} {g g' : δ → α}
    (hf : ∀ i, Measurable (f i))
    {s : Set δ} (hs : MeasurableSet s) (hg : ∀ b ∈ s, IsLUB { a | ∃ i, f i b = a } (g b))
    (hg' : EqOn g g' sᶜ) (g'_meas : Measurable g') : Measurable g 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Sort u_5} [Countable ι] {f : ι → δ → α} {g g' : δ → α} (hf : ∀ (i : ι), Measurable (f i)) {s : Set δ} (hs : MeasurableSet s) (hg : ∀ b ∈ s, IsLUB {a | ∃ i, f i b = a} (g b)) (hg' : Set.EqOn g g' sᶜ) (g'_meas : Measurable g')
Goal: Measurable g
-- AEMeasurable.isLUB
theorem AEMeasurable.isLUB {ι} {μ : Measure δ} [Countable ι] {f : ι → δ → α} {g : δ → α}
    (hf : ∀ i, AEMeasurable (f i) μ) (hg : ∀ᵐ b ∂μ, IsLUB { a | ∃ i, f i b = a } (g b)) :
    AEMeasurable g μ 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Sort u_5} {μ : MeasureTheory.Measure δ} [Countable ι] {f : ι → δ → α} {g : δ → α} (hf : ∀ (i : ι), AEMeasurable (f i) μ) (hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b))
Goal: AEMeasurable g μ
-- Measurable.isGLB
theorem Measurable.isGLB {ι} [Countable ι] {f : ι → δ → α} {g : δ → α} (hf : ∀ i, Measurable (f i))
    (hg : ∀ b, IsGLB { a | ∃ i, f i b = a } (g b)) : Measurable g 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Sort u_5} [Countable ι] {f : ι → δ → α} {g : δ → α} (hf : ∀ (i : ι), Measurable (f i)) (hg : ∀ (b : δ), IsGLB {a | ∃ i, f i b = a} (g b))
Goal: Measurable g
-- Measurable.isGLB_of_mem
theorem Measurable.isGLB_of_mem {ι} [Countable ι] {f : ι → δ → α} {g g' : δ → α}
    (hf : ∀ i, Measurable (f i))
    {s : Set δ} (hs : MeasurableSet s) (hg : ∀ b ∈ s, IsGLB { a | ∃ i, f i b = a } (g b))
    (hg' : EqOn g g' sᶜ) (g'_meas : Measurable g') : Measurable g 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Sort u_5} [Countable ι] {f : ι → δ → α} {g g' : δ → α} (hf : ∀ (i : ι), Measurable (f i)) {s : Set δ} (hs : MeasurableSet s) (hg : ∀ b ∈ s, IsGLB {a | ∃ i, f i b = a} (g b)) (hg' : Set.EqOn g g' sᶜ) (g'_meas : Measurable g')
Goal: Measurable g
-- AEMeasurable.isGLB
theorem AEMeasurable.isGLB {ι} {μ : Measure δ} [Countable ι] {f : ι → δ → α} {g : δ → α}
    (hf : ∀ i, AEMeasurable (f i) μ) (hg : ∀ᵐ b ∂μ, IsGLB { a | ∃ i, f i b = a } (g b)) :
    AEMeasurable g μ 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Sort u_5} {μ : MeasureTheory.Measure δ} [Countable ι] {f : ι → δ → α} {g : δ → α} (hf : ∀ (i : ι), AEMeasurable (f i) μ) (hg : ∀ᵐ (b : δ) ∂μ, IsGLB {a | ∃ i, f i b = a} (g b))
Goal: AEMeasurable g μ
-- Monotone.measurable
protected theorem Monotone.measurable [LinearOrder β] [OrderClosedTopology β] {f : β → α}
    (hf : Monotone f) : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] [LinearOrder β] [OrderClosedTopology β] {f : β → α} (hf : Monotone f)
Goal: Measurable f
-- aemeasurable_restrict_of_monotoneOn
theorem aemeasurable_restrict_of_monotoneOn [LinearOrder β] [OrderClosedTopology β] {μ : Measure β}
    {s : Set β} (hs : MeasurableSet s) {f : β → α} (hf : MonotoneOn f s) :
    AEMeasurable f (μ.restrict s) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] [LinearOrder β] [OrderClosedTopology β] {μ : MeasureTheory.Measure β} {s : Set β} (hs : MeasurableSet s) {f : β → α} (hf : MonotoneOn f s)
Goal: AEMeasurable f (μ.restrict s)
-- Antitone.measurable
protected theorem Antitone.measurable [LinearOrder β] [OrderClosedTopology β] {f : β → α}
    (hf : Antitone f) : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] [LinearOrder β] [OrderClosedTopology β] {f : β → α} (hf : Antitone f)
Goal: Measurable f
-- aemeasurable_restrict_of_antitoneOn
theorem aemeasurable_restrict_of_antitoneOn [LinearOrder β] [OrderClosedTopology β] {μ : Measure β}
    {s : Set β} (hs : MeasurableSet s) {f : β → α} (hf : AntitoneOn f s) :
    AEMeasurable f (μ.restrict s) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] [LinearOrder β] [OrderClosedTopology β] {μ : MeasureTheory.Measure β} {s : Set β} (hs : MeasurableSet s) {f : β → α} (hf : AntitoneOn f s)
Goal: AEMeasurable f (μ.restrict s)
-- measurableSet_of_mem_nhdsWithin_Ioi_aux
theorem measurableSet_of_mem_nhdsWithin_Ioi_aux {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x)
    (h' : ∀ x ∈ s, ∃ y, x < y) : MeasurableSet s 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x) (h' : ∀ x ∈ s, ∃ y, x < y)
Goal: MeasurableSet s
-- measurableSet_of_mem_nhdsWithin_Ioi
theorem measurableSet_of_mem_nhdsWithin_Ioi {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x) :
    MeasurableSet s 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x)
Goal: MeasurableSet s
-- measurableSet_bddAbove_range
lemma measurableSet_bddAbove_range {ι} [Countable ι] {f : ι → δ → α} (hf : ∀ i, Measurable (f i)) :
    MeasurableSet {b | BddAbove (range (fun i ↦ f i b))} 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Sort u_5} [Countable ι] {f : ι → δ → α} (hf : ∀ (i : ι), Measurable (f i))
Goal: MeasurableSet {b | BddAbove (Set.range fun i => f i b)}
-- measurableSet_bddBelow_range
lemma measurableSet_bddBelow_range {ι} [Countable ι] {f : ι → δ → α} (hf : ∀ i, Measurable (f i)) :
    MeasurableSet {b | BddBelow (range (fun i ↦ f i b))} 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [LinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Sort u_5} [Countable ι] {f : ι → δ → α} (hf : ∀ (i : ι), Measurable (f i))
Goal: MeasurableSet {b | BddBelow (Set.range fun i => f i b)}
-- Measurable.iSup_Prop
@[measurability]
theorem Measurable.iSup_Prop {α} [MeasurableSpace α] [ConditionallyCompleteLattice α]
    (p : Prop) {f : δ → α} (hf : Measurable f) : Measurable fun b => ⨆ _ : p, f b 
Arguments: {δ : Type u_4} [MeasurableSpace δ] {α : Type u_5} [MeasurableSpace α] [ConditionallyCompleteLattice α] (p : Prop) {f : δ → α} (hf : Measurable f)
Goal: Measurable fun b => ⨆ (_ : p), f b
-- Measurable.iInf_Prop
@[measurability]
theorem Measurable.iInf_Prop {α} [MeasurableSpace α] [ConditionallyCompleteLattice α]
    (p : Prop) {f : δ → α} (hf : Measurable f) : Measurable fun b => ⨅ _ : p, f b 
Arguments: {δ : Type u_4} [MeasurableSpace δ] {α : Type u_5} [MeasurableSpace α] [ConditionallyCompleteLattice α] (p : Prop) {f : δ → α} (hf : Measurable f)
Goal: Measurable fun b => ⨅ (_ : p), f b
-- measurable_iSup
@[measurability]
theorem measurable_iSup {ι} [Countable ι] {f : ι → δ → α} (hf : ∀ i, Measurable (f i)) :
    Measurable (fun b ↦ ⨆ i, f i b) 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Sort u_5} [Countable ι] {f : ι → δ → α} (hf : ∀ (i : ι), Measurable (f i))
Goal: Measurable fun b => ⨆ i, f i b
-- aemeasurable_iSup
@[measurability]
theorem aemeasurable_iSup {ι} {μ : Measure δ} [Countable ι] {f : ι → δ → α}
    (hf : ∀ i, AEMeasurable (f i) μ) : AEMeasurable (fun b => ⨆ i, f i b) μ 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Sort u_5} {μ : MeasureTheory.Measure δ} [Countable ι] {f : ι → δ → α} (hf : ∀ (i : ι), AEMeasurable (f i) μ)
Goal: AEMeasurable (fun b => ⨆ i, f i b) μ
-- measurable_iInf
@[measurability]
theorem measurable_iInf {ι} [Countable ι] {f : ι → δ → α} (hf : ∀ i, Measurable (f i)) :
    Measurable fun b => ⨅ i, f i b 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Sort u_5} [Countable ι] {f : ι → δ → α} (hf : ∀ (i : ι), Measurable (f i))
Goal: Measurable fun b => ⨅ i, f i b
-- aemeasurable_iInf
@[measurability]
theorem aemeasurable_iInf {ι} {μ : Measure δ} [Countable ι] {f : ι → δ → α}
    (hf : ∀ i, AEMeasurable (f i) μ) : AEMeasurable (fun b => ⨅ i, f i b) μ 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Sort u_5} {μ : MeasureTheory.Measure δ} [Countable ι] {f : ι → δ → α} (hf : ∀ (i : ι), AEMeasurable (f i) μ)
Goal: AEMeasurable (fun b => ⨅ i, f i b) μ
-- measurable_sSup
theorem measurable_sSup {ι} {f : ι → δ → α} {s : Set ι} (hs : s.Countable)
    (hf : ∀ i ∈ s, Measurable (f i)) :
    Measurable fun x => sSup ((fun i => f i x) '' s) 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Type u_5} {f : ι → δ → α} {s : Set ι} (hs : s.Countable) (hf : ∀ i ∈ s, Measurable (f i))
Goal: Measurable fun x => sSup ((fun i => f i x) '' s)
-- measurable_sInf
theorem measurable_sInf {ι} {f : ι → δ → α} {s : Set ι} (hs : s.Countable)
    (hf : ∀ i ∈ s, Measurable (f i)) :
    Measurable fun x => sInf ((fun i => f i x) '' s) 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Type u_5} {f : ι → δ → α} {s : Set ι} (hs : s.Countable) (hf : ∀ i ∈ s, Measurable (f i))
Goal: Measurable fun x => sInf ((fun i => f i x) '' s)
-- measurable_biSup
theorem measurable_biSup {ι} (s : Set ι) {f : ι → δ → α} (hs : s.Countable)
    (hf : ∀ i ∈ s, Measurable (f i)) : Measurable fun b => ⨆ i ∈ s, f i b 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Type u_5} (s : Set ι) {f : ι → δ → α} (hs : s.Countable) (hf : ∀ i ∈ s, Measurable (f i))
Goal: Measurable fun b => ⨆ i ∈ s, f i b
-- aemeasurable_biSup
theorem aemeasurable_biSup {ι} {μ : Measure δ} (s : Set ι) {f : ι → δ → α} (hs : s.Countable)
    (hf : ∀ i ∈ s, AEMeasurable (f i) μ) : AEMeasurable (fun b => ⨆ i ∈ s, f i b) μ 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Type u_5} {μ : MeasureTheory.Measure δ} (s : Set ι) {f : ι → δ → α} (hs : s.Countable) (hf : ∀ i ∈ s, AEMeasurable (f i) μ)
Goal: AEMeasurable (fun b => ⨆ i ∈ s, f i b) μ
-- measurable_biInf
theorem measurable_biInf {ι} (s : Set ι) {f : ι → δ → α} (hs : s.Countable)
    (hf : ∀ i ∈ s, Measurable (f i)) : Measurable fun b => ⨅ i ∈ s, f i b 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Type u_5} (s : Set ι) {f : ι → δ → α} (hs : s.Countable) (hf : ∀ i ∈ s, Measurable (f i))
Goal: Measurable fun b => ⨅ i ∈ s, f i b
-- aemeasurable_biInf
theorem aemeasurable_biInf {ι} {μ : Measure δ} (s : Set ι) {f : ι → δ → α} (hs : s.Countable)
    (hf : ∀ i ∈ s, AEMeasurable (f i) μ) : AEMeasurable (fun b => ⨅ i ∈ s, f i b) μ 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Type u_5} {μ : MeasureTheory.Measure δ} (s : Set ι) {f : ι → δ → α} (hs : s.Countable) (hf : ∀ i ∈ s, AEMeasurable (f i) μ)
Goal: AEMeasurable (fun b => ⨅ i ∈ s, f i b) μ
-- measurable_liminf'
theorem measurable_liminf' {ι ι'} {f : ι → δ → α} {v : Filter ι} (hf : ∀ i, Measurable (f i))
    {p : ι' → Prop} {s : ι' → Set ι} (hv : v.HasCountableBasis p s) (hs : ∀ j, (s j).Countable) :
    Measurable fun x => liminf (f · x) v 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Type u_5} {ι' : Type u_6} {f : ι → δ → α} {v : Filter ι} (hf : ∀ (i : ι), Measurable (f i)) {p : ι' → Prop} {s : ι' → Set ι} (hv : v.HasCountableBasis p s) (hs : ∀ (j : ι'), (s j).Countable)
Goal: Measurable fun x => Filter.liminf (fun x_1 => f x_1 x) v
-- measurable_limsup'
theorem measurable_limsup' {ι ι'} {f : ι → δ → α} {u : Filter ι} (hf : ∀ i, Measurable (f i))
    {p : ι' → Prop} {s : ι' → Set ι} (hu : u.HasCountableBasis p s) (hs : ∀ i, (s i).Countable) :
    Measurable fun x => limsup (fun i => f i x) u 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [SecondCountableTopology α] {ι : Type u_5} {ι' : Type u_6} {f : ι → δ → α} {u : Filter ι} (hf : ∀ (i : ι), Measurable (f i)) {p : ι' → Prop} {s : ι' → Set ι} (hu : u.HasCountableBasis p s) (hs : ∀ (i : ι'), (s i).Countable)
Goal: Measurable fun x => Filter.limsup (fun i => f i x) u
-- measurable_liminf
@[measurability]
theorem measurable_liminf {f : ℕ → δ → α} (hf : ∀ i, Measurable (f i)) :
    Measurable fun x => liminf (fun i => f i x) atTop 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [SecondCountableTopology α] {f : ℕ → δ → α} (hf : ∀ (i : ℕ), Measurable (f i))
Goal: Measurable fun x => Filter.liminf (fun i => f i x) Filter.atTop
-- measurable_limsup
@[measurability]
theorem measurable_limsup {f : ℕ → δ → α} (hf : ∀ i, Measurable (f i)) :
    Measurable fun x => limsup (fun i => f i x) atTop 
Arguments: {α : Type u_1} {δ : Type u_4} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [MeasurableSpace δ] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [SecondCountableTopology α] {f : ℕ → δ → α} (hf : ∀ (i : ℕ), Measurable (f i))
Goal: Measurable fun x => Filter.limsup (fun i => f i x) Filter.atTop
-- IsFiniteMeasureOnCompacts.map
protected theorem IsFiniteMeasureOnCompacts.map (μ : Measure α) [IsFiniteMeasureOnCompacts μ]
    (f : α ≃ₜ β) : IsFiniteMeasureOnCompacts (Measure.map f μ) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasureOnCompacts μ] (f : α ≃ₜ β)
Goal: MeasureTheory.IsFiniteMeasureOnCompacts (MeasureTheory.Measure.map (⇑f) μ)
-- measure_eq_measure_preimage_add_measure_tsum_Ico_zpow
theorem measure_eq_measure_preimage_add_measure_tsum_Ico_zpow {α : Type*} [MeasurableSpace α]
    (μ : Measure α) {f : α → ℝ≥0∞} (hf : Measurable f) {s : Set α} (hs : MeasurableSet s)
    {t : ℝ≥0} (ht : 1 < t) :
    μ s =
      μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' {∞}) +
      ∑' n : ℤ, μ (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) 
Arguments: {α : Type u_5} [MeasurableSpace α] (μ : MeasureTheory.Measure α) {f : α → ℝ≥0∞} (hf : Measurable f) {s : Set α} (hs : MeasurableSet s) {t : ℝ≥0} (ht : 1 < t)
Goal: μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' {⊤}) + ∑' (n : ℤ), μ (s ∩ f ⁻¹' Set.Ico (↑t ^ n) (↑t ^ (n + 1)))
-- MeasureTheory.Lp.coeFn_le
theorem coeFn_le (f g : Lp E p μ) : f ≤ᵐ[μ] g ↔ f ≤ g 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [NormedLatticeAddCommGroup E] (f g : ↥(MeasureTheory.Lp E p μ))
Goal: ↑↑f ≤ᵐ[μ] ↑↑g ↔ f ≤ g
-- MeasureTheory.Lp.coeFn_nonneg
theorem coeFn_nonneg (f : Lp E p μ) : 0 ≤ᵐ[μ] f ↔ 0 ≤ f 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [NormedLatticeAddCommGroup E] (f : ↥(MeasureTheory.Lp E p μ))
Goal: 0 ≤ᵐ[μ] ↑↑f ↔ 0 ≤ f
-- MeasureTheory.Memℒp.sup
theorem _root_.MeasureTheory.Memℒp.sup {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) :
    Memℒp (f ⊔ g) p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [NormedLatticeAddCommGroup E] {f g : α → E} (hf : MeasureTheory.Memℒp f p μ) (hg : MeasureTheory.Memℒp g p μ)
Goal: MeasureTheory.Memℒp (f ⊔ g) p μ
-- MeasureTheory.Memℒp.inf
theorem _root_.MeasureTheory.Memℒp.inf {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) :
    Memℒp (f ⊓ g) p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [NormedLatticeAddCommGroup E] {f g : α → E} (hf : MeasureTheory.Memℒp f p μ) (hg : MeasureTheory.Memℒp g p μ)
Goal: MeasureTheory.Memℒp (f ⊓ g) p μ
-- MeasureTheory.Memℒp.abs
theorem _root_.MeasureTheory.Memℒp.abs {f : α → E} (hf : Memℒp f p μ) : Memℒp |f| p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [NormedLatticeAddCommGroup E] {f : α → E} (hf : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.Memℒp |f| p μ
-- MeasureTheory.Lp.coeFn_sup
theorem coeFn_sup (f g : Lp E p μ) : ⇑(f ⊔ g) =ᵐ[μ] ⇑f ⊔ ⇑g 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [NormedLatticeAddCommGroup E] (f g : ↥(MeasureTheory.Lp E p μ))
Goal: ↑↑(f ⊔ g) =ᵐ[μ] ↑↑f ⊔ ↑↑g
-- MeasureTheory.Lp.coeFn_inf
theorem coeFn_inf (f g : Lp E p μ) : ⇑(f ⊓ g) =ᵐ[μ] ⇑f ⊓ ⇑g 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [NormedLatticeAddCommGroup E] (f g : ↥(MeasureTheory.Lp E p μ))
Goal: ↑↑(f ⊓ g) =ᵐ[μ] ↑↑f ⊓ ↑↑g
-- MeasureTheory.Lp.coeFn_abs
theorem coeFn_abs (f : Lp E p μ) : ⇑|f| =ᵐ[μ] fun x => |f x| 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [NormedLatticeAddCommGroup E] (f : ↥(MeasureTheory.Lp E p μ))
Goal: ↑↑|f| =ᵐ[μ] fun x => |↑↑f x|
-- IsPiSystem.pi
theorem IsPiSystem.pi {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i)) :
    IsPiSystem (pi univ '' pi univ C) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} {C : (i : ι) → Set (Set (α i))} (hC : ∀ (i : ι), IsPiSystem (C i))
Goal: IsPiSystem (Set.univ.pi '' Set.univ.pi C)
-- isPiSystem_pi
theorem isPiSystem_pi [∀ i, MeasurableSpace (α i)] :
    IsPiSystem (pi univ '' pi univ fun i => { s : Set (α i) | MeasurableSet s }) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [(i : ι) → MeasurableSpace (α i)]
Goal: IsPiSystem (Set.univ.pi '' Set.univ.pi fun i => {s | MeasurableSet s})
-- IsCountablySpanning.pi
theorem IsCountablySpanning.pi {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsCountablySpanning (C i)) :
    IsCountablySpanning (pi univ '' pi univ C) 
Arguments: {ι : Type u_1} {α : ι → Type u_3} [Finite ι] {C : (i : ι) → Set (Set (α i))} (hC : ∀ (i : ι), IsCountablySpanning (C i))
Goal: IsCountablySpanning (Set.univ.pi '' Set.univ.pi C)