ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.le_liminf_measure_open_of_forall_tendsto_measure
lemma le_liminf_measure_open_of_forall_tendsto_measure
    {Ω ι : Type*} {L : Filter ι} [NeBot L]
    [MeasurableSpace Ω] [PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω]
    {μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
    (h : ∀ {E}, MeasurableSet E → μ (frontier E) = 0 → Tendsto (fun i ↦ μs i E) L (𝓝 (μ E)))
    (G : Set Ω) (G_open : IsOpen G) :
    μ G ≤ L.liminf (fun i ↦ μs i G) 
Arguments: {Ω : Type u_2} {ι : Type u_3} {L : Filter ι} [L.NeBot] [MeasurableSpace Ω] [PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {μs : ι → MeasureTheory.Measure Ω} [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μs i)] (h : ∀ {E : Set Ω}, MeasurableSet E → μ (frontier E) = 0 → Filter.Tendsto (fun i => (μs i) E) L (𝓝 (μ E))) (G : Set Ω) (G_open : IsOpen G)
Goal: μ G ≤ Filter.liminf (fun i => (μs i) G) L
-- MeasureTheory.lintegral_le_liminf_lintegral_of_forall_isOpen_measure_le_liminf_measure
lemma lintegral_le_liminf_lintegral_of_forall_isOpen_measure_le_liminf_measure
    {μ : Measure Ω} {μs : ℕ → Measure Ω} {f : Ω → ℝ} (f_cont : Continuous f) (f_nn : 0 ≤ f)
    (h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
    ∫⁻ x, ENNReal.ofReal (f x) ∂μ ≤ atTop.liminf (fun i ↦ ∫⁻ x, ENNReal.ofReal (f x) ∂ (μs i)) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {μs : ℕ → MeasureTheory.Measure Ω} {f : Ω → ℝ} (f_cont : Continuous f) (f_nn : 0 ≤ f) (h_opens : ∀ (G : Set Ω), IsOpen G → μ G ≤ Filter.liminf (fun i => (μs i) G) Filter.atTop)
Goal: ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ Filter.liminf (fun i => ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) Filter.atTop
-- MeasureTheory.integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
    {μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
    {f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
    (h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
    ∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {μs : ℕ → MeasureTheory.Measure Ω} [∀ (i : ℕ), MeasureTheory.IsProbabilityMeasure (μs i)] {f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f) (h_opens : ∀ (G : Set Ω), IsOpen G → μ G ≤ Filter.liminf (fun i => (μs i) G) Filter.atTop)
Goal: ∫ (x : Ω), f x ∂μ ≤ Filter.liminf (fun i => ∫ (x : Ω), f x ∂μs i) Filter.atTop
-- MeasureTheory.tendsto_of_forall_isOpen_le_liminf
theorem tendsto_of_forall_isOpen_le_liminf {μ : ProbabilityMeasure Ω}
    {μs : ℕ → ProbabilityMeasure Ω}
    (h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
    atTop.Tendsto (fun i ↦ μs i) (𝓝 μ) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] {μ : MeasureTheory.ProbabilityMeasure Ω} {μs : ℕ → MeasureTheory.ProbabilityMeasure Ω} (h_opens : ∀ (G : Set Ω), IsOpen G → μ G ≤ Filter.liminf (fun i => (μs i) G) Filter.atTop)
Goal: Filter.Tendsto (fun i => μs i) Filter.atTop (𝓝 μ)
-- MeasureTheory.trim_eq_self
@[simp]
theorem trim_eq_self [MeasurableSpace α] {μ : Measure α} : μ.trim le_rfl = μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α}
Goal: μ.trim ⋯ = μ
-- MeasureTheory.toOuterMeasure_trim_eq_trim_toOuterMeasure
theorem toOuterMeasure_trim_eq_trim_toOuterMeasure (μ : Measure α) (hm : m ≤ m0) :
    @Measure.toOuterMeasure _ m (μ.trim hm) = @OuterMeasure.trim _ m μ.toOuterMeasure 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) (hm : m ≤ m0)
Goal: (μ.trim hm).toOuterMeasure = μ.trim
-- MeasureTheory.zero_trim
@[simp]
theorem zero_trim (hm : m ≤ m0) : (0 : Measure α).trim hm = (0 : @Measure α m) 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} (hm : m ≤ m0)
Goal: MeasureTheory.Measure.trim 0 hm = 0
-- MeasureTheory.trim_measurableSet_eq
theorem trim_measurableSet_eq (hm : m ≤ m0) (hs : @MeasurableSet α m s) : μ.trim hm s = μ s 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hm : m ≤ m0) (hs : MeasurableSet s)
Goal: (μ.trim hm) s = μ s
-- MeasureTheory.le_trim
theorem le_trim (hm : m ≤ m0) : μ s ≤ μ.trim hm s 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hm : m ≤ m0)
Goal: μ s ≤ (μ.trim hm) s
-- MeasureTheory.trim_add
lemma trim_add {ν : Measure α} (hm : m ≤ m0) : (μ + ν).trim hm = μ.trim hm + ν.trim hm 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} (hm : m ≤ m0)
Goal: (μ + ν).trim hm = μ.trim hm + ν.trim hm
-- MeasureTheory.measure_eq_zero_of_trim_eq_zero
theorem measure_eq_zero_of_trim_eq_zero (hm : m ≤ m0) (h : μ.trim hm s = 0) : μ s = 0 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hm : m ≤ m0) (h : (μ.trim hm) s = 0)
Goal: μ s = 0
-- MeasureTheory.measure_trim_toMeasurable_eq_zero
theorem measure_trim_toMeasurable_eq_zero {hm : m ≤ m0} (hs : μ.trim hm s = 0) :
    μ (@toMeasurable α m (μ.trim hm) s) = 0 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {hm : m ≤ m0} (hs : (μ.trim hm) s = 0)
Goal: μ (MeasureTheory.toMeasurable (μ.trim hm) s) = 0
-- MeasureTheory.ae_of_ae_trim
theorem ae_of_ae_trim (hm : m ≤ m0) {μ : Measure α} {P : α → Prop} (h : ∀ᵐ x ∂μ.trim hm, P x) :
    ∀ᵐ x ∂μ, P x 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} (hm : m ≤ m0) {μ : MeasureTheory.Measure α} {P : α → Prop} (h : ∀ᵐ (x : α) ∂μ.trim hm, P x)
Goal: ∀ᵐ (x : α) ∂μ, P x
-- MeasureTheory.ae_eq_of_ae_eq_trim
theorem ae_eq_of_ae_eq_trim {E} {hm : m ≤ m0} {f₁ f₂ : α → E}
    (h12 : f₁ =ᵐ[μ.trim hm] f₂) : f₁ =ᵐ[μ] f₂ 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_2} {hm : m ≤ m0} {f₁ f₂ : α → E} (h12 : f₁ =ᵐ[μ.trim hm] f₂)
Goal: f₁ =ᵐ[μ] f₂
-- MeasureTheory.ae_le_of_ae_le_trim
theorem ae_le_of_ae_le_trim {E} [LE E] {hm : m ≤ m0} {f₁ f₂ : α → E}
    (h12 : f₁ ≤ᵐ[μ.trim hm] f₂) : f₁ ≤ᵐ[μ] f₂ 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_2} [LE E] {hm : m ≤ m0} {f₁ f₂ : α → E} (h12 : f₁ ≤ᵐ[μ.trim hm] f₂)
Goal: f₁ ≤ᵐ[μ] f₂
-- MeasureTheory.trim_trim
theorem trim_trim {m₁ m₂ : MeasurableSpace α} {hm₁₂ : m₁ ≤ m₂} {hm₂ : m₂ ≤ m0} :
    (μ.trim hm₂).trim hm₁₂ = μ.trim (hm₁₂.trans hm₂) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {m₁ m₂ : MeasurableSpace α} {hm₁₂ : m₁ ≤ m₂} {hm₂ : m₂ ≤ m0}
Goal: (μ.trim hm₂).trim hm₁₂ = μ.trim ⋯
-- MeasureTheory.restrict_trim
theorem restrict_trim (hm : m ≤ m0) (μ : Measure α) (hs : @MeasurableSet α m s) :
    @Measure.restrict α m (μ.trim hm) s = (μ.restrict s).trim hm 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {s : Set α} (hm : m ≤ m0) (μ : MeasureTheory.Measure α) (hs : MeasurableSet s)
Goal: (μ.trim hm).restrict s = (μ.restrict s).trim hm
-- MeasureTheory.sigmaFiniteTrim_mono
theorem sigmaFiniteTrim_mono {m m₂ m0 : MeasurableSpace α} {μ : Measure α} (hm : m ≤ m0)
    (hm₂ : m₂ ≤ m) [SigmaFinite (μ.trim (hm₂.trans hm))] : SigmaFinite (μ.trim hm) 
Arguments: {α : Type u_1} {m m₂ m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) (hm₂ : m₂ ≤ m) [MeasureTheory.SigmaFinite (μ.trim ⋯)]
Goal: MeasureTheory.SigmaFinite (μ.trim hm)
-- MeasureTheory.sigmaFinite_trim_bot_iff
theorem sigmaFinite_trim_bot_iff : SigmaFinite (μ.trim bot_le) ↔ IsFiniteMeasure μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α}
Goal: MeasureTheory.SigmaFinite (μ.trim ⋯) ↔ MeasureTheory.IsFiniteMeasure μ
-- MeasureTheory.Measure.AbsolutelyContinuous.trim
lemma Measure.AbsolutelyContinuous.trim {ν : Measure α} (hμν : μ ≪ ν) (hm : m ≤ m0) :
    μ.trim hm ≪ ν.trim hm 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} (hμν : μ.AbsolutelyContinuous ν) (hm : m ≤ m0)
Goal: (μ.trim hm).AbsolutelyContinuous (ν.trim hm)
-- MeasureTheory.Measure.measurable_coe
theorem measurable_coe {s : Set α} (hs : MeasurableSet s) : Measurable fun μ : Measure α => μ s 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : Set α} (hs : MeasurableSet s)
Goal: Measurable fun μ => μ s
-- MeasureTheory.Measure.measurable_of_measurable_coe
theorem measurable_of_measurable_coe (f : β → Measure α)
    (h : ∀ (s : Set α), MeasurableSet s → Measurable fun b => f b s) : Measurable f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (f : β → MeasureTheory.Measure α) (h : ∀ (s : Set α), MeasurableSet s → Measurable fun b => (f b) s)
Goal: Measurable f
-- MeasureTheory.Measure.measurable_measure
theorem measurable_measure {μ : α → Measure β} :
    Measurable μ ↔ ∀ (s : Set β), MeasurableSet s → Measurable fun b => μ b s 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : α → MeasureTheory.Measure β}
Goal: Measurable μ ↔ ∀ (s : Set β), MeasurableSet s → Measurable fun b => (μ b) s
-- MeasureTheory.Measure.measurable_map
theorem measurable_map (f : α → β) (hf : Measurable f) :
    Measurable fun μ : Measure α => map f μ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (f : α → β) (hf : Measurable f)
Goal: Measurable fun μ => MeasureTheory.Measure.map f μ
-- MeasureTheory.Measure.measurable_dirac
theorem measurable_dirac : Measurable (Measure.dirac : α → Measure α) 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: Measurable MeasureTheory.Measure.dirac
-- MeasureTheory.Measure.measurable_lintegral
theorem measurable_lintegral {f : α → ℝ≥0∞} (hf : Measurable f) :
    Measurable fun μ : Measure α => ∫⁻ x, f x ∂μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ≥0∞} (hf : Measurable f)
Goal: Measurable fun μ => ∫⁻ (x : α), f x ∂μ
-- MeasureTheory.Measure.join_apply
@[simp]
theorem join_apply {m : Measure (Measure α)} {s : Set α} (hs : MeasurableSet s) :
    join m s = ∫⁻ μ, μ s ∂m 
Arguments: {α : Type u_1} [MeasurableSpace α] {m : MeasureTheory.Measure (MeasureTheory.Measure α)} {s : Set α} (hs : MeasurableSet s)
Goal: m.join s = ∫⁻ (μ : MeasureTheory.Measure α), μ s ∂m
-- MeasureTheory.Measure.join_zero
@[simp]
theorem join_zero : (0 : Measure (Measure α)).join = 0 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: MeasureTheory.Measure.join 0 = 0
-- MeasureTheory.Measure.measurable_join
theorem measurable_join : Measurable (join : Measure (Measure α) → Measure α) 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: Measurable MeasureTheory.Measure.join
-- MeasureTheory.Measure.lintegral_join
theorem lintegral_join {m : Measure (Measure α)} {f : α → ℝ≥0∞} (hf : Measurable f) :
    ∫⁻ x, f x ∂join m = ∫⁻ μ, ∫⁻ x, f x ∂μ ∂m 
Arguments: {α : Type u_1} [MeasurableSpace α] {m : MeasureTheory.Measure (MeasureTheory.Measure α)} {f : α → ℝ≥0∞} (hf : Measurable f)
Goal: ∫⁻ (x : α), f x ∂m.join = ∫⁻ (μ : MeasureTheory.Measure α), ∫⁻ (x : α), f x ∂μ ∂m
-- MeasureTheory.Measure.bind_zero_left
@[simp]
theorem bind_zero_left (f : α → Measure β) : bind 0 f = 0 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (f : α → MeasureTheory.Measure β)
Goal: MeasureTheory.Measure.bind 0 f = 0
-- MeasureTheory.Measure.bind_zero_right
@[simp]
theorem bind_zero_right (m : Measure α) : bind m (0 : α → Measure β) = 0 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (m : MeasureTheory.Measure α)
Goal: m.bind 0 = 0
-- MeasureTheory.Measure.bind_zero_right'
@[simp]
theorem bind_zero_right' (m : Measure α) : bind m (fun _ => 0 : α → Measure β) = 0 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] (m : MeasureTheory.Measure α)
Goal: (m.bind fun x => 0) = 0
-- MeasureTheory.Measure.bind_apply
@[simp]
theorem bind_apply {m : Measure α} {f : α → Measure β} {s : Set β} (hs : MeasurableSet s)
    (hf : Measurable f) : bind m f s = ∫⁻ a, f a s ∂m 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {m : MeasureTheory.Measure α} {f : α → MeasureTheory.Measure β} {s : Set β} (hs : MeasurableSet s) (hf : Measurable f)
Goal: (m.bind f) s = ∫⁻ (a : α), (f a) s ∂m
-- MeasureTheory.Measure.measurable_bind'
theorem measurable_bind' {g : α → Measure β} (hg : Measurable g) : Measurable fun m => bind m g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {g : α → MeasureTheory.Measure β} (hg : Measurable g)
Goal: Measurable fun m => m.bind g
-- MeasureTheory.Measure.lintegral_bind
theorem lintegral_bind {m : Measure α} {μ : α → Measure β} {f : β → ℝ≥0∞} (hμ : Measurable μ)
    (hf : Measurable f) : ∫⁻ x, f x ∂bind m μ = ∫⁻ a, ∫⁻ x, f x ∂μ a ∂m 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {m : MeasureTheory.Measure α} {μ : α → MeasureTheory.Measure β} {f : β → ℝ≥0∞} (hμ : Measurable μ) (hf : Measurable f)
Goal: ∫⁻ (x : β), f x ∂m.bind μ = ∫⁻ (a : α), ∫⁻ (x : β), f x ∂μ a ∂m
-- MeasureTheory.Measure.bind_bind
theorem bind_bind {γ} [MeasurableSpace γ] {m : Measure α} {f : α → Measure β} {g : β → Measure γ}
    (hf : Measurable f) (hg : Measurable g) : bind (bind m f) g = bind m fun a => bind (f a) g 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {γ : Type u_3} [MeasurableSpace γ] {m : MeasureTheory.Measure α} {f : α → MeasureTheory.Measure β} {g : β → MeasureTheory.Measure γ} (hf : Measurable f) (hg : Measurable g)
Goal: (m.bind f).bind g = m.bind fun a => (f a).bind g
-- MeasureTheory.Measure.bind_dirac
theorem bind_dirac {f : α → Measure β} (hf : Measurable f) (a : α) : bind (dirac a) f = f a 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → MeasureTheory.Measure β} (hf : Measurable f) (a : α)
Goal: (MeasureTheory.Measure.dirac a).bind f = f a
-- MeasureTheory.Measure.dirac_bind
theorem dirac_bind {m : Measure α} : bind m dirac = m 
Arguments: {α : Type u_1} [MeasurableSpace α] {m : MeasureTheory.Measure α}
Goal: m.bind MeasureTheory.Measure.dirac = m
-- MeasureTheory.Measure.join_eq_bind
theorem join_eq_bind (μ : Measure (Measure α)) : join μ = bind μ id 
Arguments: {α : Type u_1} [MeasurableSpace α] (μ : MeasureTheory.Measure (MeasureTheory.Measure α))
Goal: μ.join = μ.bind id
-- MeasureTheory.Measure.join_map_map
theorem join_map_map {f : α → β} (hf : Measurable f) (μ : Measure (Measure α)) :
    join (map (map f) μ) = map f (join μ) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {f : α → β} (hf : Measurable f) (μ : MeasureTheory.Measure (MeasureTheory.Measure α))
Goal: (MeasureTheory.Measure.map (MeasureTheory.Measure.map f) μ).join = MeasureTheory.Measure.map f μ.join
-- MeasureTheory.Measure.join_map_join
theorem join_map_join (μ : Measure (Measure (Measure α))) : join (map join μ) = join (join μ) 
Arguments: {α : Type u_1} [MeasurableSpace α] (μ : MeasureTheory.Measure (MeasureTheory.Measure (MeasureTheory.Measure α)))
Goal: (MeasureTheory.Measure.map MeasureTheory.Measure.join μ).join = μ.join.join
-- MeasureTheory.Measure.join_map_dirac
theorem join_map_dirac (μ : Measure α) : join (map dirac μ) = μ 
Arguments: {α : Type u_1} [MeasurableSpace α] (μ : MeasureTheory.Measure α)
Goal: (MeasureTheory.Measure.map MeasureTheory.Measure.dirac μ).join = μ
-- MeasureTheory.Measure.join_dirac
theorem join_dirac (μ : Measure α) : join (dirac μ) = μ 
Arguments: {α : Type u_1} [MeasurableSpace α] (μ : MeasureTheory.Measure α)
Goal: (MeasureTheory.Measure.dirac μ).join = μ
-- MeasureTheory.condexpIndL1Fin_ae_eq_condexpIndSMul
theorem condexpIndL1Fin_ae_eq_condexpIndSMul (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
    (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : G) :
    condexpIndL1Fin hm hs hμs x =ᵐ[μ] condexpIndSMul hm hs hμs x 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [NormedSpace ℝ G] (hm : m ≤ m0) [MeasureTheory.SigmaFinite (μ.trim hm)] (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x : G)
Goal: ↑↑(MeasureTheory.condexpIndL1Fin hm hs hμs x) =ᵐ[μ] ↑↑(MeasureTheory.condexpIndSMul hm hs hμs x)
-- _private.Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1.0.MeasureTheory.q
private theorem q {hs : MeasurableSet s} {hμs : μ s ≠ ∞} {x : G} :
    Memℒp (condexpIndSMul hm hs hμs x) 1 μ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [NormedSpace ℝ G] {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] {hs : MeasurableSet s} {hμs : μ s ≠ ⊤} {x : G}
Goal: MeasureTheory.Memℒp (↑↑(MeasureTheory.condexpIndSMul hm hs hμs x)) 1 μ
-- MeasureTheory.condexpIndL1Fin_add
theorem condexpIndL1Fin_add (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x y : G) :
    condexpIndL1Fin hm hs hμs (x + y) =
    condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm hs hμs y 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [NormedSpace ℝ G] {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x y : G)
Goal: MeasureTheory.condexpIndL1Fin hm hs hμs (x + y) =
  MeasureTheory.condexpIndL1Fin hm hs hμs x + MeasureTheory.condexpIndL1Fin hm hs hμs y
-- MeasureTheory.condexpIndL1Fin_smul
theorem condexpIndL1Fin_smul (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (c : ℝ) (x : G) :
    condexpIndL1Fin hm hs hμs (c • x) = c • condexpIndL1Fin hm hs hμs x 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [NormedSpace ℝ G] {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (c : ℝ) (x : G)
Goal: MeasureTheory.condexpIndL1Fin hm hs hμs (c • x) = c • MeasureTheory.condexpIndL1Fin hm hs hμs x
-- MeasureTheory.condexpIndL1Fin_smul'
theorem condexpIndL1Fin_smul' [NormedSpace ℝ F] [SMulCommClass ℝ 𝕜 F] (hs : MeasurableSet s)
    (hμs : μ s ≠ ∞) (c : 𝕜) (x : F) :
    condexpIndL1Fin hm hs hμs (c • x) = c • condexpIndL1Fin hm hs hμs x 
Arguments: {α : Type u_1} {F : Type u_3} {𝕜 : Type u_7} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] [NormedSpace ℝ F] [SMulCommClass ℝ 𝕜 F] (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (c : 𝕜) (x : F)
Goal: MeasureTheory.condexpIndL1Fin hm hs hμs (c • x) = c • MeasureTheory.condexpIndL1Fin hm hs hμs x
-- MeasureTheory.norm_condexpIndL1Fin_le
theorem norm_condexpIndL1Fin_le (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : G) :
    ‖condexpIndL1Fin hm hs hμs x‖ ≤ (μ s).toReal * ‖x‖ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [NormedSpace ℝ G] {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x : G)
Goal: ‖MeasureTheory.condexpIndL1Fin hm hs hμs x‖ ≤ (μ s).toReal * ‖x‖
-- MeasureTheory.condexpIndL1Fin_disjoint_union
theorem condexpIndL1Fin_disjoint_union (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
    (hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (x : G) :
    condexpIndL1Fin hm (hs.union ht) ((measure_union_le s t).trans_lt
      (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne x =
    condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm ht hμt x 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} [NormedSpace ℝ G] {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ⊤) (hμt : μ t ≠ ⊤) (hst : s ∩ t = ∅) (x : G)
Goal: MeasureTheory.condexpIndL1Fin hm ⋯ ⋯ x =
  MeasureTheory.condexpIndL1Fin hm hs hμs x + MeasureTheory.condexpIndL1Fin hm ht hμt x
-- MeasureTheory.condexpIndL1_of_measurableSet_of_measure_ne_top
theorem condexpIndL1_of_measurableSet_of_measure_ne_top (hs : MeasurableSet s) (hμs : μ s ≠ ∞)
    (x : G) : condexpIndL1 hm μ s x = condexpIndL1Fin hm hs hμs x 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [NormedSpace ℝ G] {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x : G)
Goal: MeasureTheory.condexpIndL1 hm μ s x = MeasureTheory.condexpIndL1Fin hm hs hμs x
-- MeasureTheory.condexpIndL1_of_measure_eq_top
theorem condexpIndL1_of_measure_eq_top (hμs : μ s = ∞) (x : G) : condexpIndL1 hm μ s x = 0 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [NormedSpace ℝ G] {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (hμs : μ s = ⊤) (x : G)
Goal: MeasureTheory.condexpIndL1 hm μ s x = 0
-- MeasureTheory.condexpIndL1_of_not_measurableSet
theorem condexpIndL1_of_not_measurableSet (hs : ¬MeasurableSet s) (x : G) :
    condexpIndL1 hm μ s x = 0 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [NormedSpace ℝ G] {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (hs : ¬MeasurableSet s) (x : G)
Goal: MeasureTheory.condexpIndL1 hm μ s x = 0
-- MeasureTheory.condexpIndL1_add
theorem condexpIndL1_add (x y : G) :
    condexpIndL1 hm μ s (x + y) = condexpIndL1 hm μ s x + condexpIndL1 hm μ s y 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [NormedSpace ℝ G] {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (x y : G)
Goal: MeasureTheory.condexpIndL1 hm μ s (x + y) = MeasureTheory.condexpIndL1 hm μ s x + MeasureTheory.condexpIndL1 hm μ s y
-- MeasureTheory.condexpIndL1_smul
theorem condexpIndL1_smul (c : ℝ) (x : G) :
    condexpIndL1 hm μ s (c • x) = c • condexpIndL1 hm μ s x 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [NormedSpace ℝ G] {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (c : ℝ) (x : G)
Goal: MeasureTheory.condexpIndL1 hm μ s (c • x) = c • MeasureTheory.condexpIndL1 hm μ s x
-- MeasureTheory.condexpIndL1_smul'
theorem condexpIndL1_smul' [NormedSpace ℝ F] [SMulCommClass ℝ 𝕜 F] (c : 𝕜) (x : F) :
    condexpIndL1 hm μ s (c • x) = c • condexpIndL1 hm μ s x 
Arguments: {α : Type u_1} {F : Type u_3} {𝕜 : Type u_7} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] [NormedSpace ℝ F] [SMulCommClass ℝ 𝕜 F] (c : 𝕜) (x : F)
Goal: MeasureTheory.condexpIndL1 hm μ s (c • x) = c • MeasureTheory.condexpIndL1 hm μ s x
-- MeasureTheory.norm_condexpIndL1_le
theorem norm_condexpIndL1_le (x : G) : ‖condexpIndL1 hm μ s x‖ ≤ (μ s).toReal * ‖x‖ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [NormedSpace ℝ G] {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)] (x : G)
Goal: ‖MeasureTheory.condexpIndL1 hm μ s x‖ ≤ (μ s).toReal * ‖x‖
-- MeasureTheory.continuous_condexpIndL1
theorem continuous_condexpIndL1 : Continuous fun x : G => condexpIndL1 hm μ s x 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [NormedSpace ℝ G] {hm : m ≤ m0} [MeasureTheory.SigmaFinite (μ.trim hm)]
Goal: Continuous fun x => MeasureTheory.condexpIndL1 hm μ s x