ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Measure.IsEverywherePos.smul_measure_nnreal
lemma IsEverywherePos.smul_measure_nnreal (hs : IsEverywherePos μ s) {c : ℝ≥0} (hc : c ≠ 0) :
    IsEverywherePos (c • μ) s 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (hs : μ.IsEverywherePos s) {c : ℝ≥0} (hc : c ≠ 0)
Goal: (c • μ).IsEverywherePos s
-- MeasureTheory.Measure.IsEverywherePos.of_forall_exists_nhds_eq
lemma IsEverywherePos.of_forall_exists_nhds_eq (hs : IsEverywherePos μ s)
    (h : ∀ x ∈ s, ∃ t ∈ 𝓝 x, ∀ u ⊆ t, ν u = μ u) : IsEverywherePos ν s 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] {μ ν : MeasureTheory.Measure α} {s : Set α} (hs : μ.IsEverywherePos s) (h : ∀ x ∈ s, ∃ t ∈ 𝓝 x, ∀ u ⊆ t, ν u = μ u)
Goal: ν.IsEverywherePos s
-- MeasureTheory.Measure.isEverywherePos_iff_of_forall_exists_nhds_eq
lemma isEverywherePos_iff_of_forall_exists_nhds_eq (h : ∀ x ∈ s, ∃ t ∈ 𝓝 x, ∀ u ⊆ t, ν u = μ u) :
    IsEverywherePos ν s ↔ IsEverywherePos μ s 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] {μ ν : MeasureTheory.Measure α} {s : Set α} (h : ∀ x ∈ s, ∃ t ∈ 𝓝 x, ∀ u ⊆ t, ν u = μ u)
Goal: ν.IsEverywherePos s ↔ μ.IsEverywherePos s
-- IsOpen.isEverywherePos
lemma _root_.IsOpen.isEverywherePos [IsOpenPosMeasure μ] (hs : IsOpen s) : IsEverywherePos μ s 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} [μ.IsOpenPosMeasure] (hs : IsOpen s)
Goal: μ.IsEverywherePos s
-- MeasureTheory.Measure.IsEverywherePos.IsGdelta_of_isMulLeftInvariant
@[to_additive]
lemma IsEverywherePos.IsGdelta_of_isMulLeftInvariant
    {k : Set G} (h : μ.IsEverywherePos k) (hk : IsCompact k) (h'k : IsClosed k) :
    IsGδ k 
Arguments: {G : Type u_2} [Group G] [TopologicalSpace G] [TopologicalGroup G] [LocallyCompactSpace G] [MeasurableSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [μ.IsMulLeftInvariant] [MeasureTheory.IsFiniteMeasureOnCompacts μ] [μ.InnerRegularCompactLTTop] {k : Set G} (h : μ.IsEverywherePos k) (hk : IsCompact k) (h'k : IsClosed k)
Goal: IsGδ k
-- MeasureTheory.Measure.IsEverywherePos.IsGdelta_of_isAddLeftInvariant
@[to_additive]
lemma IsEverywherePos.IsGdelta_of_isMulLeftInvariant
    {k : Set G} (h : μ.IsEverywherePos k) (hk : IsCompact k) (h'k : IsClosed k) :
    IsGδ k 
Arguments: {G : Type u_2} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] [LocallyCompactSpace G] [MeasurableSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [μ.IsAddLeftInvariant] [MeasureTheory.IsFiniteMeasureOnCompacts μ] [μ.InnerRegularCompactLTTop] {k : Set G} (h : μ.IsEverywherePos k) (hk : IsCompact k) (h'k : IsClosed k)
Goal: IsGδ k
-- MeasureTheory.Measure.innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_group
@[to_additive innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_addGroup]
theorem innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_group :
    InnerRegularWRT μ (fun s ↦ ∃ (f : G → ℝ), Continuous f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1})
    (fun s ↦ MeasurableSet s ∧ μ s ≠ ∞) 
Arguments: {G : Type u_2} [Group G] [TopologicalSpace G] [TopologicalGroup G] [LocallyCompactSpace G] [MeasurableSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [μ.IsMulLeftInvariant] [MeasureTheory.IsFiniteMeasureOnCompacts μ] [μ.InnerRegularCompactLTTop]
Goal: μ.InnerRegularWRT (fun s => ∃ f, Continuous f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1}) fun s => MeasurableSet s ∧ μ s ≠ ⊤
-- MeasureTheory.Measure.innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_addGroup
@[to_additive innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_addGroup]
theorem innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_group :
    InnerRegularWRT μ (fun s ↦ ∃ (f : G → ℝ), Continuous f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1})
    (fun s ↦ MeasurableSet s ∧ μ s ≠ ∞) 
Arguments: {G : Type u_2} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] [LocallyCompactSpace G] [MeasurableSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [μ.IsAddLeftInvariant] [MeasureTheory.IsFiniteMeasureOnCompacts μ] [μ.InnerRegularCompactLTTop]
Goal: μ.InnerRegularWRT (fun s => ∃ f, Continuous f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1}) fun s => MeasurableSet s ∧ μ s ≠ ⊤
-- LinearMap.exists_map_addHaar_eq_smul_addHaar'
theorem LinearMap.exists_map_addHaar_eq_smul_addHaar' (h : Function.Surjective L) :
    ∃ (c : ℝ≥0∞), 0 < c ∧ c < ∞ ∧ μ.map L = (c * addHaar (univ : Set (LinearMap.ker L))) • ν 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [NormedAddCommGroup E] [MeasurableSpace E] [BorelSpace E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [MeasurableSpace F] [BorelSpace F] [NormedSpace 𝕜 F] (L : E →ₗ[𝕜] F) (μ : MeasureTheory.Measure E) (ν : MeasureTheory.Measure F) [μ.IsAddHaarMeasure] [ν.IsAddHaarMeasure] [LocallyCompactSpace E] (h : Function.Surjective ⇑L)
Goal: ∃ c, 0 < c ∧ c < ⊤ ∧ MeasureTheory.Measure.map (⇑L) μ = (c * MeasureTheory.Measure.addHaar Set.univ) • ν
-- LinearMap.exists_map_addHaar_eq_smul_addHaar
theorem LinearMap.exists_map_addHaar_eq_smul_addHaar (h : Function.Surjective L) :
    ∃ (c : ℝ≥0∞), 0 < c ∧ μ.map L = c • ν 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [NormedAddCommGroup E] [MeasurableSpace E] [BorelSpace E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [MeasurableSpace F] [BorelSpace F] [NormedSpace 𝕜 F] (L : E →ₗ[𝕜] F) (μ : MeasureTheory.Measure E) (ν : MeasureTheory.Measure F) [μ.IsAddHaarMeasure] [ν.IsAddHaarMeasure] [LocallyCompactSpace E] (h : Function.Surjective ⇑L)
Goal: ∃ c, 0 < c ∧ MeasureTheory.Measure.map (⇑L) μ = c • ν
-- MeasureTheory.ae_comp_linearMap_mem_iff
lemma ae_comp_linearMap_mem_iff (h : Function.Surjective L) {s : Set F} (hs : MeasurableSet s) :
    (∀ᵐ x ∂μ, L x ∈ s) ↔ ∀ᵐ y ∂ν, y ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [NormedAddCommGroup E] [MeasurableSpace E] [BorelSpace E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [MeasurableSpace F] [BorelSpace F] [NormedSpace 𝕜 F] (L : E →ₗ[𝕜] F) (μ : MeasureTheory.Measure E) (ν : MeasureTheory.Measure F) [μ.IsAddHaarMeasure] [ν.IsAddHaarMeasure] [LocallyCompactSpace E] (h : Function.Surjective ⇑L) {s : Set F} (hs : MeasurableSet s)
Goal: (∀ᵐ (x : E) ∂μ, L x ∈ s) ↔ ∀ᵐ (y : F) ∂ν, y ∈ s
-- MeasureTheory.ae_ae_add_linearMap_mem_iff
lemma ae_ae_add_linearMap_mem_iff [LocallyCompactSpace F] {s : Set F} (hs : MeasurableSet s) :
    (∀ᵐ y ∂ν, ∀ᵐ x ∂μ, y + L x ∈ s) ↔ ∀ᵐ y ∂ν, y ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [NormedAddCommGroup E] [MeasurableSpace E] [BorelSpace E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [MeasurableSpace F] [BorelSpace F] [NormedSpace 𝕜 F] (L : E →ₗ[𝕜] F) (μ : MeasureTheory.Measure E) (ν : MeasureTheory.Measure F) [μ.IsAddHaarMeasure] [ν.IsAddHaarMeasure] [LocallyCompactSpace E] [LocallyCompactSpace F] {s : Set F} (hs : MeasurableSet s)
Goal: (∀ᵐ (y : F) ∂ν, ∀ᵐ (x : E) ∂μ, y + L x ∈ s) ↔ ∀ᵐ (y : F) ∂ν, y ∈ s
-- MeasureTheory.ae_mem_of_ae_add_linearMap_mem
lemma ae_mem_of_ae_add_linearMap_mem [LocallyCompactSpace F] {s : Set F} (hs : MeasurableSet s)
    (h : ∀ y, ∀ᵐ x ∂μ, y + L x ∈ s) : ∀ᵐ y ∂ν, y ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [NormedAddCommGroup E] [MeasurableSpace E] [BorelSpace E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [MeasurableSpace F] [BorelSpace F] [NormedSpace 𝕜 F] (L : E →ₗ[𝕜] F) (μ : MeasureTheory.Measure E) (ν : MeasureTheory.Measure F) [μ.IsAddHaarMeasure] [ν.IsAddHaarMeasure] [LocallyCompactSpace E] [LocallyCompactSpace F] {s : Set F} (hs : MeasurableSet s) (h : ∀ (y : F), ∀ᵐ (x : E) ∂μ, y + L x ∈ s)
Goal: ∀ᵐ (y : F) ∂ν, y ∈ s
-- MeasureTheory.AEStronglyMeasurable'.congr
theorem congr (hf : AEStronglyMeasurable' m f μ) (hfg : f =ᵐ[μ] g) :
    AEStronglyMeasurable' m g μ 
Arguments: {α : Type u_1} {β : Type u_2} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f g : α → β} (hf : MeasureTheory.AEStronglyMeasurable' m f μ) (hfg : f =ᵐ[μ] g)
Goal: MeasureTheory.AEStronglyMeasurable' m g μ
-- MeasureTheory.AEStronglyMeasurable'.mono
theorem mono {m'} (hf : AEStronglyMeasurable' m f μ) (hm : m ≤ m') :
    AEStronglyMeasurable' m' f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} {m' : MeasurableSpace α} (hf : MeasureTheory.AEStronglyMeasurable' m f μ) (hm : m ≤ m')
Goal: MeasureTheory.AEStronglyMeasurable' m' f μ
-- MeasureTheory.AEStronglyMeasurable'.add
theorem add [Add β] [ContinuousAdd β] (hf : AEStronglyMeasurable' m f μ)
    (hg : AEStronglyMeasurable' m g μ) : AEStronglyMeasurable' m (f + g) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f g : α → β} [Add β] [ContinuousAdd β] (hf : MeasureTheory.AEStronglyMeasurable' m f μ) (hg : MeasureTheory.AEStronglyMeasurable' m g μ)
Goal: MeasureTheory.AEStronglyMeasurable' m (f + g) μ
-- MeasureTheory.AEStronglyMeasurable'.neg
theorem neg [AddGroup β] [TopologicalAddGroup β] {f : α → β} (hfm : AEStronglyMeasurable' m f μ) :
    AEStronglyMeasurable' m (-f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] [AddGroup β] [TopologicalAddGroup β] {f : α → β} (hfm : MeasureTheory.AEStronglyMeasurable' m f μ)
Goal: MeasureTheory.AEStronglyMeasurable' m (-f) μ
-- MeasureTheory.AEStronglyMeasurable'.sub
theorem sub [AddGroup β] [TopologicalAddGroup β] {f g : α → β} (hfm : AEStronglyMeasurable' m f μ)
    (hgm : AEStronglyMeasurable' m g μ) : AEStronglyMeasurable' m (f - g) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] [AddGroup β] [TopologicalAddGroup β] {f g : α → β} (hfm : MeasureTheory.AEStronglyMeasurable' m f μ) (hgm : MeasureTheory.AEStronglyMeasurable' m g μ)
Goal: MeasureTheory.AEStronglyMeasurable' m (f - g) μ
-- MeasureTheory.AEStronglyMeasurable'.const_smul
theorem const_smul [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (c : 𝕜) (hf : AEStronglyMeasurable' m f μ) :
    AEStronglyMeasurable' m (c • f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {𝕜 : Type u_3} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (c : 𝕜) (hf : MeasureTheory.AEStronglyMeasurable' m f μ)
Goal: MeasureTheory.AEStronglyMeasurable' m (c • f) μ
-- MeasureTheory.AEStronglyMeasurable'.const_inner
theorem const_inner {𝕜 β} [RCLike 𝕜] [NormedAddCommGroup β] [InnerProductSpace 𝕜 β] {f : α → β}
    (hfm : AEStronglyMeasurable' m f μ) (c : β) :
    AEStronglyMeasurable' m (fun x => (inner c (f x) : 𝕜)) μ 
Arguments: {α : Type u_1} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {𝕜 : Type u_4} {β : Type u_5} [RCLike 𝕜] [NormedAddCommGroup β] [InnerProductSpace 𝕜 β] {f : α → β} (hfm : MeasureTheory.AEStronglyMeasurable' m f μ) (c : β)
Goal: MeasureTheory.AEStronglyMeasurable' m (fun x => ⟪c, f x⟫_𝕜) μ
-- MeasureTheory.AEStronglyMeasurable'.stronglyMeasurable_mk
theorem stronglyMeasurable_mk {f : α → β} (hfm : AEStronglyMeasurable' m f μ) :
    StronglyMeasurable[m] (hfm.mk f) 
Arguments: {α : Type u_1} {β : Type u_2} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} (hfm : MeasureTheory.AEStronglyMeasurable' m f μ)
Goal: MeasureTheory.StronglyMeasurable (MeasureTheory.AEStronglyMeasurable'.mk f hfm)
-- MeasureTheory.AEStronglyMeasurable'.ae_eq_mk
theorem ae_eq_mk {f : α → β} (hfm : AEStronglyMeasurable' m f μ) : f =ᵐ[μ] hfm.mk f 
Arguments: {α : Type u_1} {β : Type u_2} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} (hfm : MeasureTheory.AEStronglyMeasurable' m f μ)
Goal: f =ᵐ[μ] MeasureTheory.AEStronglyMeasurable'.mk f hfm
-- MeasureTheory.AEStronglyMeasurable'.continuous_comp
theorem continuous_comp {γ} [TopologicalSpace γ] {f : α → β} {g : β → γ} (hg : Continuous g)
    (hf : AEStronglyMeasurable' m f μ) : AEStronglyMeasurable' m (g ∘ f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {γ : Type u_4} [TopologicalSpace γ] {f : α → β} {g : β → γ} (hg : Continuous g) (hf : MeasureTheory.AEStronglyMeasurable' m f μ)
Goal: MeasureTheory.AEStronglyMeasurable' m (g ∘ f) μ
-- MeasureTheory.aeStronglyMeasurable'_of_aeStronglyMeasurable'_trim
theorem aeStronglyMeasurable'_of_aeStronglyMeasurable'_trim {α β} {m m0 m0' : MeasurableSpace α}
    [TopologicalSpace β] (hm0 : m0 ≤ m0') {μ : Measure α} {f : α → β}
    (hf : AEStronglyMeasurable' m f (μ.trim hm0)) : AEStronglyMeasurable' m f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m m0 m0' : MeasurableSpace α} [TopologicalSpace β] (hm0 : m0 ≤ m0') {μ : MeasureTheory.Measure α} {f : α → β} (hf : MeasureTheory.AEStronglyMeasurable' m f (μ.trim hm0))
Goal: MeasureTheory.AEStronglyMeasurable' m f μ
-- MeasureTheory.StronglyMeasurable.aeStronglyMeasurable'
theorem StronglyMeasurable.aeStronglyMeasurable' {α β} {m _ : MeasurableSpace α}
    [TopologicalSpace β] {μ : Measure α} {f : α → β} (hf : StronglyMeasurable[m] f) :
    AEStronglyMeasurable' m f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α}
Goal: ∀ {x : MeasurableSpace α} [inst : TopologicalSpace β] {μ : MeasureTheory.Measure α} {f : α → β},
  MeasureTheory.StronglyMeasurable f → MeasureTheory.AEStronglyMeasurable' m f μ
-- MeasureTheory.ae_eq_trim_iff_of_aeStronglyMeasurable'
theorem ae_eq_trim_iff_of_aeStronglyMeasurable' {α β} [TopologicalSpace β] [MetrizableSpace β]
    {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → β} (hm : m ≤ m0)
    (hfm : AEStronglyMeasurable' m f μ) (hgm : AEStronglyMeasurable' m g μ) :
    hfm.mk f =ᵐ[μ.trim hm] hgm.mk g ↔ f =ᵐ[μ] g 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace β] [TopologicalSpace.MetrizableSpace β] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → β} (hm : m ≤ m0) (hfm : MeasureTheory.AEStronglyMeasurable' m f μ) (hgm : MeasureTheory.AEStronglyMeasurable' m g μ)
Goal: MeasureTheory.AEStronglyMeasurable'.mk f hfm =ᵐ[μ.trim hm] MeasureTheory.AEStronglyMeasurable'.mk g hgm ↔ f =ᵐ[μ] g
-- MeasureTheory.AEStronglyMeasurable.comp_ae_measurable'
theorem AEStronglyMeasurable.comp_ae_measurable' {α β γ : Type*} [TopologicalSpace β]
    {mα : MeasurableSpace α} {_ : MeasurableSpace γ} {f : α → β} {μ : Measure γ} {g : γ → α}
    (hf : AEStronglyMeasurable f (μ.map g)) (hg : AEMeasurable g μ) :
    AEStronglyMeasurable' (mα.comap g) (f ∘ g) μ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace β] {mα : MeasurableSpace α}
Goal: ∀ {x : MeasurableSpace γ} {f : α → β} {μ : MeasureTheory.Measure γ} {g : γ → α},
  MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.map g μ) →
    AEMeasurable g μ → MeasureTheory.AEStronglyMeasurable' (MeasurableSpace.comap g mα) (f ∘ g) μ
-- MeasureTheory.AEStronglyMeasurable'.aeStronglyMeasurable'_of_measurableSpace_le_on
theorem AEStronglyMeasurable'.aeStronglyMeasurable'_of_measurableSpace_le_on {α E}
    {m m₂ m0 : MeasurableSpace α} {μ : Measure α} [TopologicalSpace E] [Zero E] (hm : m ≤ m0)
    {s : Set α} {f : α → E} (hs_m : MeasurableSet[m] s)
    (hs : ∀ t, MeasurableSet[m] (s ∩ t) → MeasurableSet[m₂] (s ∩ t))
    (hf : AEStronglyMeasurable' m f μ) (hf_zero : f =ᵐ[μ.restrict sᶜ] 0) :
    AEStronglyMeasurable' m₂ f μ 
Arguments: {α : Type u_1} {E : Type u_2} {m m₂ m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace E] [Zero E] (hm : m ≤ m0) {s : Set α} {f : α → E} (hs_m : MeasurableSet s) (hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)) (hf : MeasureTheory.AEStronglyMeasurable' m f μ) (hf_zero : f =ᵐ[μ.restrict sᶜ] 0)
Goal: MeasureTheory.AEStronglyMeasurable' m₂ f μ
-- MeasureTheory.mem_lpMeasSubgroup_iff_aeStronglyMeasurable'
theorem mem_lpMeasSubgroup_iff_aeStronglyMeasurable' {m m0 : MeasurableSpace α} {μ : Measure α}
    {f : Lp F p μ} : f ∈ lpMeasSubgroup F m p μ ↔ AEStronglyMeasurable' m f μ 
Arguments: {α : Type u_1} {F : Type u_3} {p : ℝ≥0∞} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ↥(MeasureTheory.Lp F p μ)}
Goal: f ∈ MeasureTheory.lpMeasSubgroup F m p μ ↔ MeasureTheory.AEStronglyMeasurable' m (↑↑f) μ
-- MeasureTheory.mem_lpMeas_iff_aeStronglyMeasurable'
theorem mem_lpMeas_iff_aeStronglyMeasurable' {m m0 : MeasurableSpace α} {μ : Measure α}
    {f : Lp F p μ} : f ∈ lpMeas F 𝕜 m p μ ↔ AEStronglyMeasurable' m f μ 
Arguments: {α : Type u_1} {F : Type u_3} {𝕜 : Type u_5} {p : ℝ≥0∞} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ↥(MeasureTheory.Lp F p μ)}
Goal: f ∈ MeasureTheory.lpMeas F 𝕜 m p μ ↔ MeasureTheory.AEStronglyMeasurable' m (↑↑f) μ
-- MeasureTheory.lpMeas.aeStronglyMeasurable'
theorem lpMeas.aeStronglyMeasurable' {m _ : MeasurableSpace α} {μ : Measure α}
    (f : lpMeas F 𝕜 m p μ) : AEStronglyMeasurable' (β := F) m f μ 
Arguments: {α : Type u_1} {F : Type u_3} {𝕜 : Type u_5} {p : ℝ≥0∞} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m : MeasurableSpace α}
Goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : ↥(MeasureTheory.lpMeas F 𝕜 m p μ)),
  MeasureTheory.AEStronglyMeasurable' m (↑↑↑f) μ
-- MeasureTheory.mem_lpMeas_self
theorem mem_lpMeas_self {m0 : MeasurableSpace α} (μ : Measure α) (f : Lp F p μ) :
    f ∈ lpMeas F 𝕜 m0 p μ 
Arguments: {α : Type u_1} {F : Type u_3} {𝕜 : Type u_5} {p : ℝ≥0∞} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : ↥(MeasureTheory.Lp F p μ))
Goal: f ∈ MeasureTheory.lpMeas F 𝕜 m0 p μ
-- MeasureTheory.lpMeasSubgroup_coe
theorem lpMeasSubgroup_coe {m _ : MeasurableSpace α} {μ : Measure α} {f : lpMeasSubgroup F m p μ} :
    (f : _ → _) = (f : Lp F p μ) 
Arguments: {α : Type u_1} {F : Type u_3} {p : ℝ≥0∞} [NormedAddCommGroup F] {m : MeasurableSpace α}
Goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ↥(MeasureTheory.lpMeasSubgroup F m p μ)}, ↑↑↑f = ↑↑↑f
-- MeasureTheory.lpMeas_coe
theorem lpMeas_coe {m _ : MeasurableSpace α} {μ : Measure α} {f : lpMeas F 𝕜 m p μ} :
    (f : _ → _) = (f : Lp F p μ) 
Arguments: {α : Type u_1} {F : Type u_3} {𝕜 : Type u_5} {p : ℝ≥0∞} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m : MeasurableSpace α}
Goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ↥(MeasureTheory.lpMeas F 𝕜 m p μ)}, ↑↑↑f = ↑↑↑f
-- MeasureTheory.mem_lpMeas_indicatorConstLp
theorem mem_lpMeas_indicatorConstLp {m m0 : MeasurableSpace α} (hm : m ≤ m0) {μ : Measure α}
    {s : Set α} (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) {c : F} :
    indicatorConstLp p (hm s hs) hμs c ∈ lpMeas F 𝕜 m p μ 
Arguments: {α : Type u_1} {F : Type u_3} {𝕜 : Type u_5} {p : ℝ≥0∞} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m m0 : MeasurableSpace α} (hm : m ≤ m0) {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) {c : F}
Goal: MeasureTheory.indicatorConstLp p ⋯ hμs c ∈ MeasureTheory.lpMeas F 𝕜 m p μ
-- MeasureTheory.memℒp_trim_of_mem_lpMeasSubgroup
theorem memℒp_trim_of_mem_lpMeasSubgroup (hm : m ≤ m0) (f : Lp F p μ)
    (hf_meas : f ∈ lpMeasSubgroup F m p μ) :
    Memℒp (mem_lpMeasSubgroup_iff_aeStronglyMeasurable'.mp hf_meas).choose p (μ.trim hm) 
Arguments: {α : Type u_1} {F : Type u_3} {p : ℝ≥0∞} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) (f : ↥(MeasureTheory.Lp F p μ)) (hf_meas : f ∈ MeasureTheory.lpMeasSubgroup F m p μ)
Goal: MeasureTheory.Memℒp (Exists.choose ⋯) p (μ.trim hm)
-- MeasureTheory.mem_lpMeasSubgroup_toLp_of_trim
theorem mem_lpMeasSubgroup_toLp_of_trim (hm : m ≤ m0) (f : Lp F p (μ.trim hm)) :
    (memℒp_of_memℒp_trim hm (Lp.memℒp f)).toLp f ∈ lpMeasSubgroup F m p μ 
Arguments: {α : Type u_1} {F : Type u_3} {p : ℝ≥0∞} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) (f : ↥(MeasureTheory.Lp F p (μ.trim hm)))
Goal: MeasureTheory.Memℒp.toLp ↑↑f ⋯ ∈ MeasureTheory.lpMeasSubgroup F m p μ
-- MeasureTheory.lpMeasSubgroupToLpTrim_ae_eq
theorem lpMeasSubgroupToLpTrim_ae_eq (hm : m ≤ m0) (f : lpMeasSubgroup F m p μ) :
    lpMeasSubgroupToLpTrim F p μ hm f =ᵐ[μ] f 
Arguments: {α : Type u_1} {F : Type u_3} {p : ℝ≥0∞} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) (f : ↥(MeasureTheory.lpMeasSubgroup F m p μ))
Goal: ↑↑(MeasureTheory.lpMeasSubgroupToLpTrim F p μ hm f) =ᵐ[μ] ↑↑↑f
-- MeasureTheory.lpTrimToLpMeasSubgroup_ae_eq
theorem lpTrimToLpMeasSubgroup_ae_eq (hm : m ≤ m0) (f : Lp F p (μ.trim hm)) :
    lpTrimToLpMeasSubgroup F p μ hm f =ᵐ[μ] f 
Arguments: {α : Type u_1} {F : Type u_3} {p : ℝ≥0∞} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) (f : ↥(MeasureTheory.Lp F p (μ.trim hm)))
Goal: ↑↑↑(MeasureTheory.lpTrimToLpMeasSubgroup F p μ hm f) =ᵐ[μ] ↑↑f
-- MeasureTheory.lpMeasToLpTrim_ae_eq
theorem lpMeasToLpTrim_ae_eq (hm : m ≤ m0) (f : lpMeas F 𝕜 m p μ) :
    lpMeasToLpTrim F 𝕜 p μ hm f =ᵐ[μ] f 
Arguments: {α : Type u_1} {F : Type u_3} {𝕜 : Type u_5} {p : ℝ≥0∞} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) (f : ↥(MeasureTheory.lpMeas F 𝕜 m p μ))
Goal: ↑↑(MeasureTheory.lpMeasToLpTrim F 𝕜 p μ hm f) =ᵐ[μ] ↑↑↑f
-- MeasureTheory.lpTrimToLpMeas_ae_eq
theorem lpTrimToLpMeas_ae_eq (hm : m ≤ m0) (f : Lp F p (μ.trim hm)) :
    lpTrimToLpMeas F 𝕜 p μ hm f =ᵐ[μ] f 
Arguments: {α : Type u_1} {F : Type u_3} {𝕜 : Type u_5} {p : ℝ≥0∞} [RCLike 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) (f : ↥(MeasureTheory.Lp F p (μ.trim hm)))
Goal: ↑↑↑(MeasureTheory.lpTrimToLpMeas F 𝕜 p μ hm f) =ᵐ[μ] ↑↑f
-- MeasureTheory.lpMeasSubgroupToLpTrim_right_inv
theorem lpMeasSubgroupToLpTrim_right_inv (hm : m ≤ m0) :
    Function.RightInverse (lpTrimToLpMeasSubgroup F p μ hm) (lpMeasSubgroupToLpTrim F p μ hm) 
Arguments: {α : Type u_1} {F : Type u_3} {p : ℝ≥0∞} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0)
Goal: Function.RightInverse (MeasureTheory.lpTrimToLpMeasSubgroup F p μ hm) (MeasureTheory.lpMeasSubgroupToLpTrim F p μ hm)
-- MeasureTheory.lpMeasSubgroupToLpTrim_left_inv
theorem lpMeasSubgroupToLpTrim_left_inv (hm : m ≤ m0) :
    Function.LeftInverse (lpTrimToLpMeasSubgroup F p μ hm) (lpMeasSubgroupToLpTrim F p μ hm) 
Arguments: {α : Type u_1} {F : Type u_3} {p : ℝ≥0∞} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0)
Goal: Function.LeftInverse (MeasureTheory.lpTrimToLpMeasSubgroup F p μ hm) (MeasureTheory.lpMeasSubgroupToLpTrim F p μ hm)
-- MeasureTheory.lpMeasSubgroupToLpTrim_add
theorem lpMeasSubgroupToLpTrim_add (hm : m ≤ m0) (f g : lpMeasSubgroup F m p μ) :
    lpMeasSubgroupToLpTrim F p μ hm (f + g) =
      lpMeasSubgroupToLpTrim F p μ hm f + lpMeasSubgroupToLpTrim F p μ hm g 
Arguments: {α : Type u_1} {F : Type u_3} {p : ℝ≥0∞} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) (f g : ↥(MeasureTheory.lpMeasSubgroup F m p μ))
Goal: MeasureTheory.lpMeasSubgroupToLpTrim F p μ hm (f + g) =
  MeasureTheory.lpMeasSubgroupToLpTrim F p μ hm f + MeasureTheory.lpMeasSubgroupToLpTrim F p μ hm g
-- MeasureTheory.lpMeasSubgroupToLpTrim_neg
theorem lpMeasSubgroupToLpTrim_neg (hm : m ≤ m0) (f : lpMeasSubgroup F m p μ) :
    lpMeasSubgroupToLpTrim F p μ hm (-f) = -lpMeasSubgroupToLpTrim F p μ hm f 
Arguments: {α : Type u_1} {F : Type u_3} {p : ℝ≥0∞} [NormedAddCommGroup F] {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (hm : m ≤ m0) (f : ↥(MeasureTheory.lpMeasSubgroup F m p μ))
Goal: MeasureTheory.lpMeasSubgroupToLpTrim F p μ hm (-f) = -MeasureTheory.lpMeasSubgroupToLpTrim F p μ hm f