ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.VectorMeasure.zero_apply
theorem zero_apply (i : Set α) : (0 : VectorMeasure α M) i = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] (i : Set α)
Goal: ↑0 i = 0
-- MeasureTheory.VectorMeasure.coe_add
@[simp]
theorem coe_add (v w : VectorMeasure α M) : ⇑(v + w) = v + w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] [ContinuousAdd M] (v w : MeasureTheory.VectorMeasure α M)
Goal: ↑(v + w) = ↑v + ↑w
-- MeasureTheory.VectorMeasure.add_apply
theorem add_apply (v w : VectorMeasure α M) (i : Set α) : (v + w) i = v i + w i 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] [ContinuousAdd M] (v w : MeasureTheory.VectorMeasure α M) (i : Set α)
Goal: ↑(v + w) i = ↑v i + ↑w i
-- MeasureTheory.VectorMeasure.coeFnAddMonoidHom_apply
@[simps]
def coeFnAddMonoidHom : VectorMeasure α M →+ Set α → M 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] [ContinuousAdd M] (self : MeasureTheory.VectorMeasure α M)
Goal: ∀ (a : Set α), MeasureTheory.VectorMeasure.coeFnAddMonoidHom self a = ↑self a
-- MeasureTheory.VectorMeasure.coe_neg
@[simp]
theorem coe_neg (v : VectorMeasure α M) : ⇑(-v) = -v 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommGroup M] [TopologicalSpace M] [TopologicalAddGroup M] (v : MeasureTheory.VectorMeasure α M)
Goal: ↑(-v) = -↑v
-- MeasureTheory.VectorMeasure.neg_apply
theorem neg_apply (v : VectorMeasure α M) (i : Set α) : (-v) i = -v i 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommGroup M] [TopologicalSpace M] [TopologicalAddGroup M] (v : MeasureTheory.VectorMeasure α M) (i : Set α)
Goal: ↑(-v) i = -↑v i
-- MeasureTheory.VectorMeasure.coe_sub
@[simp]
theorem coe_sub (v w : VectorMeasure α M) : ⇑(v - w) = v - w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommGroup M] [TopologicalSpace M] [TopologicalAddGroup M] (v w : MeasureTheory.VectorMeasure α M)
Goal: ↑(v - w) = ↑v - ↑w
-- MeasureTheory.VectorMeasure.sub_apply
theorem sub_apply (v w : VectorMeasure α M) (i : Set α) : (v - w) i = v i - w i 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommGroup M] [TopologicalSpace M] [TopologicalAddGroup M] (v w : MeasureTheory.VectorMeasure α M) (i : Set α)
Goal: ↑(v - w) i = ↑v i - ↑w i
-- MeasureTheory.Measure.toSignedMeasure_apply
@[simps]
def toSignedMeasure (μ : Measure α) [hμ : IsFiniteMeasure μ] : SignedMeasure α 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) [hμ : MeasureTheory.IsFiniteMeasure μ] (s : Set α)
Goal: ↑μ.toSignedMeasure s = if MeasurableSet s then (μ s).toReal else 0
-- MeasureTheory.Measure.toSignedMeasure_apply_measurable
theorem toSignedMeasure_apply_measurable {μ : Measure α} [IsFiniteMeasure μ] {i : Set α}
    (hi : MeasurableSet i) : μ.toSignedMeasure i = (μ i).toReal 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {i : Set α} (hi : MeasurableSet i)
Goal: ↑μ.toSignedMeasure i = (μ i).toReal
-- MeasureTheory.Measure.toSignedMeasure_congr
theorem toSignedMeasure_congr {μ ν : Measure α} [IsFiniteMeasure μ] [IsFiniteMeasure ν]
    (h : μ = ν) : μ.toSignedMeasure = ν.toSignedMeasure 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν] (h : μ = ν)
Goal: μ.toSignedMeasure = ν.toSignedMeasure
-- MeasureTheory.Measure.toSignedMeasure_eq_toSignedMeasure_iff
theorem toSignedMeasure_eq_toSignedMeasure_iff {μ ν : Measure α} [IsFiniteMeasure μ]
    [IsFiniteMeasure ν] : μ.toSignedMeasure = ν.toSignedMeasure ↔ μ = ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν]
Goal: μ.toSignedMeasure = ν.toSignedMeasure ↔ μ = ν
-- MeasureTheory.Measure.toSignedMeasure_zero
@[simp]
theorem toSignedMeasure_zero : (0 : Measure α).toSignedMeasure = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α}
Goal: MeasureTheory.Measure.toSignedMeasure 0 = 0
-- MeasureTheory.Measure.toSignedMeasure_add
@[simp]
theorem toSignedMeasure_add (μ ν : Measure α) [IsFiniteMeasure μ] [IsFiniteMeasure ν] :
    (μ + ν).toSignedMeasure = μ.toSignedMeasure + ν.toSignedMeasure 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν]
Goal: (μ + ν).toSignedMeasure = μ.toSignedMeasure + ν.toSignedMeasure
-- MeasureTheory.Measure.toSignedMeasure_smul
@[simp]
theorem toSignedMeasure_smul (μ : Measure α) [IsFiniteMeasure μ] (r : ℝ≥0) :
    (r • μ).toSignedMeasure = r • μ.toSignedMeasure 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (r : ℝ≥0)
Goal: (r • μ).toSignedMeasure = r • μ.toSignedMeasure
-- MeasureTheory.Measure.toENNRealVectorMeasure_apply
@[simps]
def toENNRealVectorMeasure (μ : Measure α) : VectorMeasure α ℝ≥0∞ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (i : Set α)
Goal: ↑μ.toENNRealVectorMeasure i = if MeasurableSet i then μ i else 0
-- MeasureTheory.Measure.toENNRealVectorMeasure_apply_measurable
theorem toENNRealVectorMeasure_apply_measurable {μ : Measure α} {i : Set α} (hi : MeasurableSet i) :
    μ.toENNRealVectorMeasure i = μ i 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {i : Set α} (hi : MeasurableSet i)
Goal: ↑μ.toENNRealVectorMeasure i = μ i
-- MeasureTheory.Measure.toENNRealVectorMeasure_zero
@[simp]
theorem toENNRealVectorMeasure_zero : (0 : Measure α).toENNRealVectorMeasure = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α}
Goal: MeasureTheory.Measure.toENNRealVectorMeasure 0 = 0
-- MeasureTheory.Measure.toENNRealVectorMeasure_add
@[simp]
theorem toENNRealVectorMeasure_add (μ ν : Measure α) :
    (μ + ν).toENNRealVectorMeasure = μ.toENNRealVectorMeasure + ν.toENNRealVectorMeasure 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α)
Goal: (μ + ν).toENNRealVectorMeasure = μ.toENNRealVectorMeasure + ν.toENNRealVectorMeasure
-- MeasureTheory.Measure.toSignedMeasure_sub_apply
theorem toSignedMeasure_sub_apply {μ ν : Measure α} [IsFiniteMeasure μ] [IsFiniteMeasure ν]
    {i : Set α} (hi : MeasurableSet i) :
    (μ.toSignedMeasure - ν.toSignedMeasure) i = (μ i).toReal - (ν i).toReal 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν] {i : Set α} (hi : MeasurableSet i)
Goal: ↑(μ.toSignedMeasure - ν.toSignedMeasure) i = (μ i).toReal - (ν i).toReal
-- MeasureTheory.VectorMeasure.ennrealToMeasure_apply
theorem ennrealToMeasure_apply {m : MeasurableSpace α} {v : VectorMeasure α ℝ≥0∞} {s : Set α}
    (hs : MeasurableSet s) : ennrealToMeasure v s = v s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {v : MeasureTheory.VectorMeasure α ℝ≥0∞} {s : Set α} (hs : MeasurableSet s)
Goal: v.ennrealToMeasure s = ↑v s
-- MeasureTheory.Measure.toENNRealVectorMeasure_ennrealToMeasure
@[simp]
theorem _root_.MeasureTheory.Measure.toENNRealVectorMeasure_ennrealToMeasure
    (μ : VectorMeasure α ℝ≥0∞) :
    toENNRealVectorMeasure (ennrealToMeasure μ) = μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.VectorMeasure α ℝ≥0∞)
Goal: μ.ennrealToMeasure.toENNRealVectorMeasure = μ
-- MeasureTheory.VectorMeasure.ennrealToMeasure_toENNRealVectorMeasure
@[simp]
theorem ennrealToMeasure_toENNRealVectorMeasure (μ : Measure α) :
    ennrealToMeasure (toENNRealVectorMeasure μ) = μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α)
Goal: μ.toENNRealVectorMeasure.ennrealToMeasure = μ
-- MeasureTheory.VectorMeasure.equivMeasure_symm_apply
@[simps]
def equivMeasure [MeasurableSpace α] : VectorMeasure α ℝ≥0∞ ≃ Measure α 
Arguments: {α : Type u_1} [MeasurableSpace α] (μ : MeasureTheory.Measure α)
Goal: MeasureTheory.VectorMeasure.equivMeasure.symm μ = μ.toENNRealVectorMeasure
-- MeasureTheory.VectorMeasure.equivMeasure_apply
@[simps]
def equivMeasure [MeasurableSpace α] : VectorMeasure α ℝ≥0∞ ≃ Measure α 
Arguments: {α : Type u_1} [MeasurableSpace α] (v : MeasureTheory.VectorMeasure α ℝ≥0∞)
Goal: MeasureTheory.VectorMeasure.equivMeasure v = v.ennrealToMeasure
-- MeasureTheory.VectorMeasure.map_not_measurable
theorem map_not_measurable {f : α → β} (hf : ¬Measurable f) : v.map f = 0 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] (v : MeasureTheory.VectorMeasure α M) {f : α → β} (hf : ¬Measurable f)
Goal: v.map f = 0
-- MeasureTheory.VectorMeasure.map_apply
theorem map_apply {f : α → β} (hf : Measurable f) {s : Set β} (hs : MeasurableSet s) :
    v.map f s = v (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] (v : MeasureTheory.VectorMeasure α M) {f : α → β} (hf : Measurable f) {s : Set β} (hs : MeasurableSet s)
Goal: ↑(v.map f) s = ↑v (f ⁻¹' s)
-- MeasureTheory.VectorMeasure.map_id
@[simp]
theorem map_id : v.map id = v 
Arguments: {α : Type u_1} [MeasurableSpace α] {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] (v : MeasureTheory.VectorMeasure α M)
Goal: v.map id = v
-- MeasureTheory.VectorMeasure.map_zero
@[simp]
theorem map_zero (f : α → β) : (0 : VectorMeasure α M).map f = 0 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] (f : α → β)
Goal: MeasureTheory.VectorMeasure.map 0 f = 0
-- MeasureTheory.VectorMeasure.mapRange_apply
@[simp]
theorem mapRange_apply {f : M →+ N} (hf : Continuous f) {s : Set α} : v.mapRange f hf s = f (v s) 
Arguments: {α : Type u_1} [MeasurableSpace α] {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] (v : MeasureTheory.VectorMeasure α M) {N : Type u_4} [AddCommMonoid N] [TopologicalSpace N] {f : M →+ N} (hf : Continuous ⇑f) {s : Set α}
Goal: ↑(v.mapRange f hf) s = f (↑v s)
-- MeasureTheory.VectorMeasure.mapRange_id
@[simp]
theorem mapRange_id : v.mapRange (AddMonoidHom.id M) continuous_id = v 
Arguments: {α : Type u_1} [MeasurableSpace α] {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] (v : MeasureTheory.VectorMeasure α M)
Goal: v.mapRange (AddMonoidHom.id M) ⋯ = v
-- MeasureTheory.VectorMeasure.mapRange_zero
@[simp]
theorem mapRange_zero {f : M →+ N} (hf : Continuous f) :
    mapRange (0 : VectorMeasure α M) f hf = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] {N : Type u_4} [AddCommMonoid N] [TopologicalSpace N] {f : M →+ N} (hf : Continuous ⇑f)
Goal: MeasureTheory.VectorMeasure.mapRange 0 f hf = 0
-- MeasureTheory.VectorMeasure.mapRange_add
@[simp]
theorem mapRange_add {v w : VectorMeasure α M} {f : M →+ N} (hf : Continuous f) :
    (v + w).mapRange f hf = v.mapRange f hf + w.mapRange f hf 
Arguments: {α : Type u_1} [MeasurableSpace α] {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] {N : Type u_4} [AddCommMonoid N] [TopologicalSpace N] [ContinuousAdd M] [ContinuousAdd N] {v w : MeasureTheory.VectorMeasure α M} {f : M →+ N} (hf : Continuous ⇑f)
Goal: (v + w).mapRange f hf = v.mapRange f hf + w.mapRange f hf
-- MeasureTheory.VectorMeasure.restrict_not_measurable
theorem restrict_not_measurable {i : Set α} (hi : ¬MeasurableSet i) : v.restrict i = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] (v : MeasureTheory.VectorMeasure α M) {i : Set α} (hi : ¬MeasurableSet i)
Goal: v.restrict i = 0
-- MeasureTheory.VectorMeasure.restrict_apply
theorem restrict_apply {i : Set α} (hi : MeasurableSet i) {j : Set α} (hj : MeasurableSet j) :
    v.restrict i j = v (j ∩ i) 
Arguments: {α : Type u_1} [MeasurableSpace α] {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] (v : MeasureTheory.VectorMeasure α M) {i : Set α} (hi : MeasurableSet i) {j : Set α} (hj : MeasurableSet j)
Goal: ↑(v.restrict i) j = ↑v (j ∩ i)
-- MeasureTheory.VectorMeasure.restrict_eq_self
theorem restrict_eq_self {i : Set α} (hi : MeasurableSet i) {j : Set α} (hj : MeasurableSet j)
    (hij : j ⊆ i) : v.restrict i j = v j 
Arguments: {α : Type u_1} [MeasurableSpace α] {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] (v : MeasureTheory.VectorMeasure α M) {i : Set α} (hi : MeasurableSet i) {j : Set α} (hj : MeasurableSet j) (hij : j ⊆ i)
Goal: ↑(v.restrict i) j = ↑v j
-- MeasureTheory.VectorMeasure.restrict_empty
@[simp]
theorem restrict_empty : v.restrict ∅ = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] (v : MeasureTheory.VectorMeasure α M)
Goal: v.restrict ∅ = 0
-- MeasureTheory.VectorMeasure.restrict_univ
@[simp]
theorem restrict_univ : v.restrict Set.univ = v 
Arguments: {α : Type u_1} [MeasurableSpace α] {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] (v : MeasureTheory.VectorMeasure α M)
Goal: v.restrict Set.univ = v
-- MeasureTheory.VectorMeasure.restrict_zero
@[simp]
theorem restrict_zero {i : Set α} : (0 : VectorMeasure α M).restrict i = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] {i : Set α}
Goal: MeasureTheory.VectorMeasure.restrict 0 i = 0
-- MeasureTheory.VectorMeasure.map_add
theorem map_add (v w : VectorMeasure α M) (f : α → β) : (v + w).map f = v.map f + w.map f 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] [ContinuousAdd M] (v w : MeasureTheory.VectorMeasure α M) (f : α → β)
Goal: (v + w).map f = v.map f + w.map f
-- MeasureTheory.VectorMeasure.mapGm_apply
@[simps]
def mapGm (f : α → β) : VectorMeasure α M →+ VectorMeasure β M 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] [ContinuousAdd M] (f : α → β) (v : MeasureTheory.VectorMeasure α M)
Goal: (MeasureTheory.VectorMeasure.mapGm f) v = v.map f
-- MeasureTheory.VectorMeasure.restrict_add
theorem restrict_add (v w : VectorMeasure α M) (i : Set α) :
    (v + w).restrict i = v.restrict i + w.restrict i 
Arguments: {α : Type u_1} [MeasurableSpace α] {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] [ContinuousAdd M] (v w : MeasureTheory.VectorMeasure α M) (i : Set α)
Goal: (v + w).restrict i = v.restrict i + w.restrict i
-- MeasureTheory.VectorMeasure.restrictGm_apply
@[simps]
def restrictGm (i : Set α) : VectorMeasure α M →+ VectorMeasure α M 
Arguments: {α : Type u_1} [MeasurableSpace α] {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] [ContinuousAdd M] (i : Set α) (v : MeasureTheory.VectorMeasure α M)
Goal: (MeasureTheory.VectorMeasure.restrictGm i) v = v.restrict i
-- MeasureTheory.VectorMeasure.map_smul
@[simp]
theorem map_smul {v : VectorMeasure α M} {f : α → β} (c : R) : (c • v).map f = c • v.map f 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [MeasurableSpace β] {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] {R : Type u_4} [Semiring R] [DistribMulAction R M] [ContinuousConstSMul R M] {v : MeasureTheory.VectorMeasure α M} {f : α → β} (c : R)
Goal: (c • v).map f = c • v.map f
-- MeasureTheory.VectorMeasure.restrict_smul
@[simp]
theorem restrict_smul {v : VectorMeasure α M} {i : Set α} (c : R) :
    (c • v).restrict i = c • v.restrict i 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] {R : Type u_4} [Semiring R] [DistribMulAction R M] [ContinuousConstSMul R M] {v : MeasureTheory.VectorMeasure α M} {i : Set α} (c : R)
Goal: (c • v).restrict i = c • v.restrict i
-- MeasureTheory.VectorMeasure.mapₗ_apply
@[simps]
def mapₗ (f : α → β) : VectorMeasure α M →ₗ[R] VectorMeasure β M 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} [MeasurableSpace β] {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] {R : Type u_4} [Semiring R] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] (f : α → β) (v : MeasureTheory.VectorMeasure α M)
Goal: (MeasureTheory.VectorMeasure.mapₗ f) v = v.map f
-- MeasureTheory.VectorMeasure.restrictₗ_apply
@[simps]
def restrictₗ (i : Set α) : VectorMeasure α M →ₗ[R] VectorMeasure α M 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [AddCommMonoid M] [TopologicalSpace M] {R : Type u_4} [Semiring R] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] (i : Set α) (v : MeasureTheory.VectorMeasure α M)
Goal: (MeasureTheory.VectorMeasure.restrictₗ i) v = v.restrict i
-- MeasureTheory.VectorMeasure.le_iff
theorem le_iff : v ≤ w ↔ ∀ i, MeasurableSet i → v i ≤ w i 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [TopologicalSpace M] [AddCommMonoid M] [PartialOrder M] {v w : MeasureTheory.VectorMeasure α M}
Goal: v ≤ w ↔ ∀ (i : Set α), MeasurableSet i → ↑v i ≤ ↑w i
-- MeasureTheory.VectorMeasure.le_iff'
theorem le_iff' : v ≤ w ↔ ∀ i, v i ≤ w i 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [TopologicalSpace M] [AddCommMonoid M] [PartialOrder M] {v w : MeasureTheory.VectorMeasure α M}
Goal: v ≤ w ↔ ∀ (i : Set α), ↑v i ≤ ↑w i
-- MeasureTheory.VectorMeasure.restrict_le_restrict_iff
theorem restrict_le_restrict_iff {i : Set α} (hi : MeasurableSet i) :
    v ≤[i] w ↔ ∀ ⦃j⦄, MeasurableSet j → j ⊆ i → v j ≤ w j 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [TopologicalSpace M] [AddCommMonoid M] [PartialOrder M] (v w : MeasureTheory.VectorMeasure α M) {i : Set α} (hi : MeasurableSet i)
Goal: v.restrict i ≤ w.restrict i ↔ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑v j ≤ ↑w j
-- MeasureTheory.VectorMeasure.subset_le_of_restrict_le_restrict
theorem subset_le_of_restrict_le_restrict {i : Set α} (hi : MeasurableSet i) (hi₂ : v ≤[i] w)
    {j : Set α} (hj : j ⊆ i) : v j ≤ w j 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [TopologicalSpace M] [AddCommMonoid M] [PartialOrder M] (v w : MeasureTheory.VectorMeasure α M) {i : Set α} (hi : MeasurableSet i) (hi₂ : v.restrict i ≤ w.restrict i) {j : Set α} (hj : j ⊆ i)
Goal: ↑v j ≤ ↑w j
-- MeasureTheory.VectorMeasure.restrict_le_restrict_of_subset_le
theorem restrict_le_restrict_of_subset_le {i : Set α}
    (h : ∀ ⦃j⦄, MeasurableSet j → j ⊆ i → v j ≤ w j) : v ≤[i] w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [TopologicalSpace M] [AddCommMonoid M] [PartialOrder M] (v w : MeasureTheory.VectorMeasure α M) {i : Set α} (h : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑v j ≤ ↑w j)
Goal: v.restrict i ≤ w.restrict i
-- MeasureTheory.VectorMeasure.restrict_le_restrict_subset
theorem restrict_le_restrict_subset {i j : Set α} (hi₁ : MeasurableSet i) (hi₂ : v ≤[i] w)
    (hij : j ⊆ i) : v ≤[j] w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [TopologicalSpace M] [AddCommMonoid M] [PartialOrder M] (v w : MeasureTheory.VectorMeasure α M) {i j : Set α} (hi₁ : MeasurableSet i) (hi₂ : v.restrict i ≤ w.restrict i) (hij : j ⊆ i)
Goal: v.restrict j ≤ w.restrict j
-- MeasureTheory.VectorMeasure.le_restrict_empty
theorem le_restrict_empty : v ≤[∅] w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [TopologicalSpace M] [AddCommMonoid M] [PartialOrder M] (v w : MeasureTheory.VectorMeasure α M)
Goal: v.restrict ∅ ≤ w.restrict ∅
-- MeasureTheory.VectorMeasure.le_restrict_univ_iff_le
theorem le_restrict_univ_iff_le : v ≤[Set.univ] w ↔ v ≤ w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [TopologicalSpace M] [AddCommMonoid M] [PartialOrder M] (v w : MeasureTheory.VectorMeasure α M)
Goal: v.restrict Set.univ ≤ w.restrict Set.univ ↔ v ≤ w
-- MeasureTheory.VectorMeasure.neg_le_neg
nonrec theorem neg_le_neg {i : Set α} (hi : MeasurableSet i) (h : v ≤[i] w) : -w ≤[i] -v 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [TopologicalSpace M] [OrderedAddCommGroup M] [TopologicalAddGroup M] (v w : MeasureTheory.VectorMeasure α M) {i : Set α} (hi : MeasurableSet i) (h : v.restrict i ≤ w.restrict i)
Goal: (-w).restrict i ≤ (-v).restrict i
-- MeasureTheory.VectorMeasure.neg_le_neg_iff
@[simp]
theorem neg_le_neg_iff {i : Set α} (hi : MeasurableSet i) : -w ≤[i] -v ↔ v ≤[i] w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [TopologicalSpace M] [OrderedAddCommGroup M] [TopologicalAddGroup M] (v w : MeasureTheory.VectorMeasure α M) {i : Set α} (hi : MeasurableSet i)
Goal: (-w).restrict i ≤ (-v).restrict i ↔ v.restrict i ≤ w.restrict i
-- MeasureTheory.VectorMeasure.restrict_le_restrict_iUnion
theorem restrict_le_restrict_iUnion {f : ℕ → Set α} (hf₁ : ∀ n, MeasurableSet (f n))
    (hf₂ : ∀ n, v ≤[f n] w) : v ≤[⋃ n, f n] w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [TopologicalSpace M] [OrderedAddCommMonoid M] [OrderClosedTopology M] (v w : MeasureTheory.VectorMeasure α M) {f : ℕ → Set α} (hf₁ : ∀ (n : ℕ), MeasurableSet (f n)) (hf₂ : ∀ (n : ℕ), v.restrict (f n) ≤ w.restrict (f n))
Goal: v.restrict (⋃ n, f n) ≤ w.restrict (⋃ n, f n)
-- MeasureTheory.VectorMeasure.restrict_le_restrict_countable_iUnion
theorem restrict_le_restrict_countable_iUnion [Countable β] {f : β → Set α}
    (hf₁ : ∀ b, MeasurableSet (f b)) (hf₂ : ∀ b, v ≤[f b] w) : v ≤[⋃ b, f b] w 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {M : Type u_3} [TopologicalSpace M] [OrderedAddCommMonoid M] [OrderClosedTopology M] (v w : MeasureTheory.VectorMeasure α M) [Countable β] {f : β → Set α} (hf₁ : ∀ (b : β), MeasurableSet (f b)) (hf₂ : ∀ (b : β), v.restrict (f b) ≤ w.restrict (f b))
Goal: v.restrict (⋃ b, f b) ≤ w.restrict (⋃ b, f b)
-- MeasureTheory.VectorMeasure.restrict_le_restrict_union
theorem restrict_le_restrict_union (hi₁ : MeasurableSet i) (hi₂ : v ≤[i] w) (hj₁ : MeasurableSet j)
    (hj₂ : v ≤[j] w) : v ≤[i ∪ j] w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [TopologicalSpace M] [OrderedAddCommMonoid M] [OrderClosedTopology M] (v w : MeasureTheory.VectorMeasure α M) {i j : Set α} (hi₁ : MeasurableSet i) (hi₂ : v.restrict i ≤ w.restrict i) (hj₁ : MeasurableSet j) (hj₂ : v.restrict j ≤ w.restrict j)
Goal: v.restrict (i ∪ j) ≤ w.restrict (i ∪ j)
-- MeasureTheory.VectorMeasure.nonneg_of_zero_le_restrict
theorem nonneg_of_zero_le_restrict (hi₂ : 0 ≤[i] v) : 0 ≤ v i 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [TopologicalSpace M] [OrderedAddCommMonoid M] (v : MeasureTheory.VectorMeasure α M) {i : Set α} (hi₂ : MeasureTheory.VectorMeasure.restrict 0 i ≤ v.restrict i)
Goal: 0 ≤ ↑v i
-- MeasureTheory.VectorMeasure.nonpos_of_restrict_le_zero
theorem nonpos_of_restrict_le_zero (hi₂ : v ≤[i] 0) : v i ≤ 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [TopologicalSpace M] [OrderedAddCommMonoid M] (v : MeasureTheory.VectorMeasure α M) {i : Set α} (hi₂ : v.restrict i ≤ MeasureTheory.VectorMeasure.restrict 0 i)
Goal: ↑v i ≤ 0
-- MeasureTheory.VectorMeasure.zero_le_restrict_not_measurable
theorem zero_le_restrict_not_measurable (hi : ¬MeasurableSet i) : 0 ≤[i] v 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [TopologicalSpace M] [OrderedAddCommMonoid M] (v : MeasureTheory.VectorMeasure α M) {i : Set α} (hi : ¬MeasurableSet i)
Goal: MeasureTheory.VectorMeasure.restrict 0 i ≤ v.restrict i
-- MeasureTheory.VectorMeasure.restrict_le_zero_of_not_measurable
theorem restrict_le_zero_of_not_measurable (hi : ¬MeasurableSet i) : v ≤[i] 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [TopologicalSpace M] [OrderedAddCommMonoid M] (v : MeasureTheory.VectorMeasure α M) {i : Set α} (hi : ¬MeasurableSet i)
Goal: v.restrict i ≤ MeasureTheory.VectorMeasure.restrict 0 i
-- MeasureTheory.VectorMeasure.measurable_of_not_zero_le_restrict
theorem measurable_of_not_zero_le_restrict (hi : ¬0 ≤[i] v) : MeasurableSet i 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [TopologicalSpace M] [OrderedAddCommMonoid M] (v : MeasureTheory.VectorMeasure α M) {i : Set α} (hi : ¬MeasureTheory.VectorMeasure.restrict 0 i ≤ v.restrict i)
Goal: MeasurableSet i