ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.measure_isOpen_pos_of_smulInvariant_of_compact_ne_zero
@[to_additive]
theorem measure_isOpen_pos_of_smulInvariant_of_compact_ne_zero (hK : IsCompact K) (hμK : μ K ≠ 0)
    (hU : IsOpen U) (hne : U.Nonempty) : 0 < μ U 
Arguments: (G : Type u) {α : Type w} {m : MeasurableSpace α} [Group G] [MulAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.SMulInvariantMeasure G α μ] [TopologicalSpace α] [ContinuousConstSMul G α] [MulAction.IsMinimal G α] {K U : Set α} (hK : IsCompact K) (hμK : μ K ≠ 0) (hU : IsOpen U) (hne : U.Nonempty)
Goal: 0 < μ U
-- MeasureTheory.isLocallyFiniteMeasure_of_smulInvariant
@[to_additive]
theorem isLocallyFiniteMeasure_of_smulInvariant (hU : IsOpen U) (hne : U.Nonempty) (hμU : μ U ≠ ∞) :
    IsLocallyFiniteMeasure μ 
Arguments: (G : Type u) {α : Type w} {m : MeasurableSpace α} [Group G] [MulAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.SMulInvariantMeasure G α μ] [TopologicalSpace α] [ContinuousConstSMul G α] [MulAction.IsMinimal G α] {U : Set α} (hU : IsOpen U) (hne : U.Nonempty) (hμU : μ U ≠ ⊤)
Goal: MeasureTheory.IsLocallyFiniteMeasure μ
-- MeasureTheory.isLocallyFiniteMeasure_of_vaddInvariant
@[to_additive]
theorem isLocallyFiniteMeasure_of_smulInvariant (hU : IsOpen U) (hne : U.Nonempty) (hμU : μ U ≠ ∞) :
    IsLocallyFiniteMeasure μ 
Arguments: (G : Type u) {α : Type w} {m : MeasurableSpace α} [AddGroup G] [AddAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.VAddInvariantMeasure G α μ] [TopologicalSpace α] [ContinuousConstVAdd G α] [AddAction.IsMinimal G α] {U : Set α} (hU : IsOpen U) (hne : U.Nonempty) (hμU : μ U ≠ ⊤)
Goal: MeasureTheory.IsLocallyFiniteMeasure μ
-- MeasureTheory.measure_isOpen_pos_of_vaddInvariant_of_ne_zero
@[to_additive]
theorem measure_isOpen_pos_of_smulInvariant_of_ne_zero (hμ : μ ≠ 0) (hU : IsOpen U)
    (hne : U.Nonempty) : 0 < μ U 
Arguments: (G : Type u) {α : Type w} {m : MeasurableSpace α} [AddGroup G] [AddAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.VAddInvariantMeasure G α μ] [TopologicalSpace α] [ContinuousConstVAdd G α] [AddAction.IsMinimal G α] {U : Set α} [μ.Regular] (hμ : μ ≠ 0) (hU : IsOpen U) (hne : U.Nonempty)
Goal: 0 < μ U
-- MeasureTheory.measure_isOpen_pos_of_smulInvariant_of_ne_zero
@[to_additive]
theorem measure_isOpen_pos_of_smulInvariant_of_ne_zero (hμ : μ ≠ 0) (hU : IsOpen U)
    (hne : U.Nonempty) : 0 < μ U 
Arguments: (G : Type u) {α : Type w} {m : MeasurableSpace α} [Group G] [MulAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.SMulInvariantMeasure G α μ] [TopologicalSpace α] [ContinuousConstSMul G α] [MulAction.IsMinimal G α] {U : Set α} [μ.Regular] (hμ : μ ≠ 0) (hU : IsOpen U) (hne : U.Nonempty)
Goal: 0 < μ U
-- MeasureTheory.measure_pos_iff_nonempty_of_smulInvariant
@[to_additive]
theorem measure_pos_iff_nonempty_of_smulInvariant (hμ : μ ≠ 0) (hU : IsOpen U) :
    0 < μ U ↔ U.Nonempty 
Arguments: (G : Type u) {α : Type w} {m : MeasurableSpace α} [Group G] [MulAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.SMulInvariantMeasure G α μ] [TopologicalSpace α] [ContinuousConstSMul G α] [MulAction.IsMinimal G α] {U : Set α} [μ.Regular] (hμ : μ ≠ 0) (hU : IsOpen U)
Goal: 0 < μ U ↔ U.Nonempty
-- MeasureTheory.measure_pos_iff_nonempty_of_vaddInvariant
@[to_additive]
theorem measure_pos_iff_nonempty_of_smulInvariant (hμ : μ ≠ 0) (hU : IsOpen U) :
    0 < μ U ↔ U.Nonempty 
Arguments: (G : Type u) {α : Type w} {m : MeasurableSpace α} [AddGroup G] [AddAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.VAddInvariantMeasure G α μ] [TopologicalSpace α] [ContinuousConstVAdd G α] [AddAction.IsMinimal G α] {U : Set α} [μ.Regular] (hμ : μ ≠ 0) (hU : IsOpen U)
Goal: 0 < μ U ↔ U.Nonempty
-- MeasureTheory.measure_eq_zero_iff_eq_empty_of_vaddInvariant
@[to_additive]
theorem measure_eq_zero_iff_eq_empty_of_smulInvariant (hμ : μ ≠ 0) (hU : IsOpen U) :
    μ U = 0 ↔ U = ∅ 
Arguments: (G : Type u) {α : Type w} {m : MeasurableSpace α} [AddGroup G] [AddAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.VAddInvariantMeasure G α μ] [TopologicalSpace α] [ContinuousConstVAdd G α] [AddAction.IsMinimal G α] {U : Set α} [μ.Regular] (hμ : μ ≠ 0) (hU : IsOpen U)
Goal: μ U = 0 ↔ U = ∅
-- MeasureTheory.measure_eq_zero_iff_eq_empty_of_smulInvariant
@[to_additive]
theorem measure_eq_zero_iff_eq_empty_of_smulInvariant (hμ : μ ≠ 0) (hU : IsOpen U) :
    μ U = 0 ↔ U = ∅ 
Arguments: (G : Type u) {α : Type w} {m : MeasurableSpace α} [Group G] [MulAction G α] {μ : MeasureTheory.Measure α} [MeasureTheory.SMulInvariantMeasure G α μ] [TopologicalSpace α] [ContinuousConstSMul G α] [MulAction.IsMinimal G α] {U : Set α} [μ.Regular] (hμ : μ ≠ 0) (hU : IsOpen U)
Goal: μ U = 0 ↔ U = ∅
-- StieltjesFunction.mono'
structure StieltjesFunction where
  toFun : ℝ → ℝ
  mono' : Monotone toFun
  right_continuous' : ∀ x, ContinuousWithinAt toFun (Ici x) x
Arguments: (self : StieltjesFunction)
Goal: Monotone ↑self
-- StieltjesFunction.right_continuous'
structure StieltjesFunction where
  toFun : ℝ → ℝ
  mono' : Monotone toFun
  right_continuous' : ∀ x, ContinuousWithinAt toFun (Ici x) x
Arguments: (self : StieltjesFunction) (x : ℝ)
Goal: ContinuousWithinAt (↑self) (Set.Ici x) x
-- StieltjesFunction.ext
@[ext] lemma ext {f g : StieltjesFunction} (h : ∀ x, f x = g x) : f = g 
Arguments: {f g : StieltjesFunction} (h : ∀ (x : ℝ), ↑f x = ↑g x)
Goal: f = g
-- StieltjesFunction.mono
theorem mono : Monotone f 
Arguments: (f : StieltjesFunction)
Goal: Monotone ↑f
-- StieltjesFunction.right_continuous
theorem right_continuous (x : ℝ) : ContinuousWithinAt f (Ici x) x 
Arguments: (f : StieltjesFunction) (x : ℝ)
Goal: ContinuousWithinAt (↑f) (Set.Ici x) x
-- StieltjesFunction.rightLim_eq
theorem rightLim_eq (f : StieltjesFunction) (x : ℝ) : Function.rightLim f x = f x 
Arguments: (f : StieltjesFunction) (x : ℝ)
Goal: Function.rightLim (↑f) x = ↑f x
-- StieltjesFunction.iInf_Ioi_eq
theorem iInf_Ioi_eq (f : StieltjesFunction) (x : ℝ) : ⨅ r : Ioi x, f r = f x 
Arguments: (f : StieltjesFunction) (x : ℝ)
Goal: ⨅ r, ↑f ↑r = ↑f x
-- StieltjesFunction.iInf_rat_gt_eq
theorem iInf_rat_gt_eq (f : StieltjesFunction) (x : ℝ) :
    ⨅ r : { r' : ℚ // x < r' }, f r = f x 
Arguments: (f : StieltjesFunction) (x : ℝ)
Goal: ⨅ r, ↑f ↑↑r = ↑f x
-- StieltjesFunction.id_apply
@[simps]
protected def id : StieltjesFunction 
Arguments: (a : ℝ)
Goal: ↑StieltjesFunction.id a = id a
-- StieltjesFunction.id_leftLim
@[simp]
theorem id_leftLim (x : ℝ) : leftLim StieltjesFunction.id x = x 
Arguments: (x : ℝ)
Goal: Function.leftLim (↑StieltjesFunction.id) x = x
-- StieltjesFunction.const_apply
@[simp] lemma const_apply (c x : ℝ) : (StieltjesFunction.const c) x = c 
Arguments: (c x : ℝ)
Goal: ↑(StieltjesFunction.const c) x = c
-- StieltjesFunction.add_apply
@[simp] lemma add_apply (f g : StieltjesFunction) (x : ℝ) : (f + g) x = f x + g x 
Arguments: (f g : StieltjesFunction) (x : ℝ)
Goal: ↑(f + g) x = ↑f x + ↑g x
-- Monotone.stieltjesFunction_eq
theorem _root_.Monotone.stieltjesFunction_eq {f : ℝ → ℝ} (hf : Monotone f) (x : ℝ) :
    hf.stieltjesFunction x = rightLim f x 
Arguments: {f : ℝ → ℝ} (hf : Monotone f) (x : ℝ)
Goal: ↑hf.stieltjesFunction x = Function.rightLim f x
-- StieltjesFunction.countable_leftLim_ne
theorem countable_leftLim_ne (f : StieltjesFunction) : Set.Countable { x | leftLim f x ≠ f x } 
Arguments: (f : StieltjesFunction)
Goal: {x | Function.leftLim (↑f) x ≠ ↑f x}.Countable
-- StieltjesFunction.length_empty
@[simp]
theorem length_empty : f.length ∅ = 0 
Arguments: (f : StieltjesFunction)
Goal: f.length ∅ = 0
-- StieltjesFunction.length_Ioc
@[simp]
theorem length_Ioc (a b : ℝ) : f.length (Ioc a b) = ofReal (f b - f a) 
Arguments: (f : StieltjesFunction) (a b : ℝ)
Goal: f.length (Set.Ioc a b) = ENNReal.ofReal (↑f b - ↑f a)
-- StieltjesFunction.length_mono
theorem length_mono {s₁ s₂ : Set ℝ} (h : s₁ ⊆ s₂) : f.length s₁ ≤ f.length s₂ 
Arguments: (f : StieltjesFunction) {s₁ s₂ : Set ℝ} (h : s₁ ⊆ s₂)
Goal: f.length s₁ ≤ f.length s₂
-- StieltjesFunction.outer_le_length
theorem outer_le_length (s : Set ℝ) : f.outer s ≤ f.length s 
Arguments: (f : StieltjesFunction) (s : Set ℝ)
Goal: f.outer s ≤ f.length s
-- StieltjesFunction.length_subadditive_Icc_Ioo
theorem length_subadditive_Icc_Ioo {a b : ℝ} {c d : ℕ → ℝ} (ss : Icc a b ⊆ ⋃ i, Ioo (c i) (d i)) :
    ofReal (f b - f a) ≤ ∑' i, ofReal (f (d i) - f (c i)) 
Arguments: (f : StieltjesFunction) {a b : ℝ} {c d : ℕ → ℝ} (ss : Set.Icc a b ⊆ ⋃ i, Set.Ioo (c i) (d i))
Goal: ENNReal.ofReal (↑f b - ↑f a) ≤ ∑' (i : ℕ), ENNReal.ofReal (↑f (d i) - ↑f (c i))
-- StieltjesFunction.outer_Ioc
@[simp]
theorem outer_Ioc (a b : ℝ) : f.outer (Ioc a b) = ofReal (f b - f a) 
Arguments: (f : StieltjesFunction) (a b : ℝ)
Goal: f.outer (Set.Ioc a b) = ENNReal.ofReal (↑f b - ↑f a)
-- StieltjesFunction.measurableSet_Ioi
theorem measurableSet_Ioi {c : ℝ} : MeasurableSet[f.outer.caratheodory] (Ioi c) 
Arguments: (f : StieltjesFunction) {c : ℝ}
Goal: MeasurableSet (Set.Ioi c)
-- StieltjesFunction.outer_trim
theorem outer_trim : f.outer.trim = f.outer 
Arguments: (f : StieltjesFunction)
Goal: f.outer.trim = f.outer
-- StieltjesFunction.borel_le_measurable
theorem borel_le_measurable : borel ℝ ≤ f.outer.caratheodory 
Arguments: (f : StieltjesFunction)
Goal: borel ℝ ≤ f.outer.caratheodory
-- StieltjesFunction.measure_def
protected irreducible_def measure : Measure ℝ 
Arguments: (f : StieltjesFunction)
Goal: f.measure = { toOuterMeasure := f.outer, m_iUnion := ⋯, trim_le := ⋯ }
-- StieltjesFunction.measure_Ioc
@[simp]
theorem measure_Ioc (a b : ℝ) : f.measure (Ioc a b) = ofReal (f b - f a) 
Arguments: (f : StieltjesFunction) (a b : ℝ)
Goal: f.measure (Set.Ioc a b) = ENNReal.ofReal (↑f b - ↑f a)
-- StieltjesFunction.measure_singleton
@[simp]
theorem measure_singleton (a : ℝ) : f.measure {a} = ofReal (f a - leftLim f a) 
Arguments: (f : StieltjesFunction) (a : ℝ)
Goal: f.measure {a} = ENNReal.ofReal (↑f a - Function.leftLim (↑f) a)
-- StieltjesFunction.measure_Icc
@[simp]
theorem measure_Icc (a b : ℝ) : f.measure (Icc a b) = ofReal (f b - leftLim f a) 
Arguments: (f : StieltjesFunction) (a b : ℝ)
Goal: f.measure (Set.Icc a b) = ENNReal.ofReal (↑f b - Function.leftLim (↑f) a)
-- StieltjesFunction.measure_Ioo
@[simp]
theorem measure_Ioo {a b : ℝ} : f.measure (Ioo a b) = ofReal (leftLim f b - f a) 
Arguments: (f : StieltjesFunction) {a b : ℝ}
Goal: f.measure (Set.Ioo a b) = ENNReal.ofReal (Function.leftLim (↑f) b - ↑f a)
-- StieltjesFunction.measure_Ico
@[simp]
theorem measure_Ico (a b : ℝ) : f.measure (Ico a b) = ofReal (leftLim f b - leftLim f a) 
Arguments: (f : StieltjesFunction) (a b : ℝ)
Goal: f.measure (Set.Ico a b) = ENNReal.ofReal (Function.leftLim (↑f) b - Function.leftLim (↑f) a)
-- StieltjesFunction.measure_Iic
theorem measure_Iic {l : ℝ} (hf : Tendsto f atBot (𝓝 l)) (x : ℝ) :
    f.measure (Iic x) = ofReal (f x - l) 
Arguments: (f : StieltjesFunction) {l : ℝ} (hf : Filter.Tendsto (↑f) Filter.atBot (𝓝 l)) (x : ℝ)
Goal: f.measure (Set.Iic x) = ENNReal.ofReal (↑f x - l)
-- StieltjesFunction.measure_Ici
theorem measure_Ici {l : ℝ} (hf : Tendsto f atTop (𝓝 l)) (x : ℝ) :
    f.measure (Ici x) = ofReal (l - leftLim f x) 
Arguments: (f : StieltjesFunction) {l : ℝ} (hf : Filter.Tendsto (↑f) Filter.atTop (𝓝 l)) (x : ℝ)
Goal: f.measure (Set.Ici x) = ENNReal.ofReal (l - Function.leftLim (↑f) x)
-- StieltjesFunction.measure_univ
theorem measure_univ {l u : ℝ} (hfl : Tendsto f atBot (𝓝 l)) (hfu : Tendsto f atTop (𝓝 u)) :
    f.measure univ = ofReal (u - l) 
Arguments: (f : StieltjesFunction) {l u : ℝ} (hfl : Filter.Tendsto (↑f) Filter.atBot (𝓝 l)) (hfu : Filter.Tendsto (↑f) Filter.atTop (𝓝 u))
Goal: f.measure Set.univ = ENNReal.ofReal (u - l)
-- StieltjesFunction.isFiniteMeasure
lemma isFiniteMeasure {l u : ℝ} (hfl : Tendsto f atBot (𝓝 l)) (hfu : Tendsto f atTop (𝓝 u)) :
    IsFiniteMeasure f.measure 
Arguments: (f : StieltjesFunction) {l u : ℝ} (hfl : Filter.Tendsto (↑f) Filter.atBot (𝓝 l)) (hfu : Filter.Tendsto (↑f) Filter.atTop (𝓝 u))
Goal: MeasureTheory.IsFiniteMeasure f.measure
-- StieltjesFunction.isProbabilityMeasure
lemma isProbabilityMeasure (hf_bot : Tendsto f atBot (𝓝 0)) (hf_top : Tendsto f atTop (𝓝 1)) :
    IsProbabilityMeasure f.measure 
Arguments: (f : StieltjesFunction) (hf_bot : Filter.Tendsto (↑f) Filter.atBot (𝓝 0)) (hf_top : Filter.Tendsto (↑f) Filter.atTop (𝓝 1))
Goal: MeasureTheory.IsProbabilityMeasure f.measure
-- StieltjesFunction.eq_of_measure_of_tendsto_atBot
lemma eq_of_measure_of_tendsto_atBot (g : StieltjesFunction) {l : ℝ}
    (hfg : f.measure = g.measure) (hfl : Tendsto f atBot (𝓝 l)) (hgl : Tendsto g atBot (𝓝 l)) :
    f = g 
Arguments: (f g : StieltjesFunction) {l : ℝ} (hfg : f.measure = g.measure) (hfl : Filter.Tendsto (↑f) Filter.atBot (𝓝 l)) (hgl : Filter.Tendsto (↑g) Filter.atBot (𝓝 l))
Goal: f = g
-- StieltjesFunction.eq_of_measure_of_eq
lemma eq_of_measure_of_eq (g : StieltjesFunction) {y : ℝ}
    (hfg : f.measure = g.measure) (hy : f y = g y) :
    f = g 
Arguments: (f g : StieltjesFunction) {y : ℝ} (hfg : f.measure = g.measure) (hy : ↑f y = ↑g y)
Goal: f = g
-- StieltjesFunction.measure_const
@[simp]
lemma measure_const (c : ℝ) : (StieltjesFunction.const c).measure = 0 
Arguments: (c : ℝ)
Goal: (StieltjesFunction.const c).measure = 0
-- StieltjesFunction.measure_add
@[simp]
lemma measure_add (f g : StieltjesFunction) : (f + g).measure = f.measure + g.measure 
Arguments: (f g : StieltjesFunction)
Goal: (f + g).measure = f.measure + g.measure
-- StieltjesFunction.measure_smul
@[simp]
lemma measure_smul (c : ℝ≥0) (f : StieltjesFunction) : (c • f).measure = c • f.measure 
Arguments: (c : ℝ≥0) (f : StieltjesFunction)
Goal: (c • f).measure = c • f.measure
-- MeasureTheory.Measure.dirac_one_mconv
@[to_additive (attr := simp)]
theorem dirac_one_mconv [MeasurableMul₂ M] (μ : Measure M) [SFinite μ] :
    (Measure.dirac 1) ∗ μ = μ 
Arguments: {M : Type u_1} [Monoid M] [MeasurableSpace M] [MeasurableMul₂ M] (μ : MeasureTheory.Measure M) [MeasureTheory.SFinite μ]
Goal: (MeasureTheory.Measure.dirac 1).mconv μ = μ
-- MeasureTheory.Measure.dirac_zero_mconv
@[to_additive (attr := simp)]
theorem dirac_one_mconv [MeasurableMul₂ M] (μ : Measure M) [SFinite μ] :
    (Measure.dirac 1) ∗ μ = μ 
Arguments: {M : Type u_1} [AddMonoid M] [MeasurableSpace M] [MeasurableAdd₂ M] (μ : MeasureTheory.Measure M) [MeasureTheory.SFinite μ]
Goal: (MeasureTheory.Measure.dirac 0).conv μ = μ
-- MeasureTheory.Measure.mconv_dirac_zero
@[to_additive (attr := simp)]
theorem mconv_dirac_one [MeasurableMul₂ M]
    (μ : Measure M) [SFinite μ] : μ ∗ (Measure.dirac 1) = μ 
Arguments: {M : Type u_1} [AddMonoid M] [MeasurableSpace M] [MeasurableAdd₂ M] (μ : MeasureTheory.Measure M) [MeasureTheory.SFinite μ]
Goal: μ.conv (MeasureTheory.Measure.dirac 0) = μ
-- MeasureTheory.Measure.mconv_dirac_one
@[to_additive (attr := simp)]
theorem mconv_dirac_one [MeasurableMul₂ M]
    (μ : Measure M) [SFinite μ] : μ ∗ (Measure.dirac 1) = μ 
Arguments: {M : Type u_1} [Monoid M] [MeasurableSpace M] [MeasurableMul₂ M] (μ : MeasureTheory.Measure M) [MeasureTheory.SFinite μ]
Goal: μ.mconv (MeasureTheory.Measure.dirac 1) = μ
-- MeasureTheory.Measure.mconv_zero
@[to_additive (attr := simp) conv_zero]
theorem mconv_zero (μ : Measure M) : (0 : Measure M) ∗ μ = (0 : Measure M) 
Arguments: {M : Type u_1} [Monoid M] [MeasurableSpace M] (μ : MeasureTheory.Measure M)
Goal: MeasureTheory.Measure.mconv 0 μ = 0
-- MeasureTheory.Measure.conv_zero
@[to_additive (attr := simp) conv_zero]
theorem mconv_zero (μ : Measure M) : (0 : Measure M) ∗ μ = (0 : Measure M) 
Arguments: {M : Type u_1} [AddMonoid M] [MeasurableSpace M] (μ : MeasureTheory.Measure M)
Goal: MeasureTheory.Measure.conv 0 μ = 0
-- MeasureTheory.Measure.zero_conv
@[to_additive (attr := simp) zero_conv]
theorem zero_mconv (μ : Measure M) : μ ∗ (0 : Measure M) = (0 : Measure M) 
Arguments: {M : Type u_1} [AddMonoid M] [MeasurableSpace M] (μ : MeasureTheory.Measure M)
Goal: μ.conv 0 = 0
-- MeasureTheory.Measure.zero_mconv
@[to_additive (attr := simp) zero_conv]
theorem zero_mconv (μ : Measure M) : μ ∗ (0 : Measure M) = (0 : Measure M) 
Arguments: {M : Type u_1} [Monoid M] [MeasurableSpace M] (μ : MeasureTheory.Measure M)
Goal: μ.mconv 0 = 0
-- MeasureTheory.Measure.mconv_add
@[to_additive conv_add]
theorem mconv_add [MeasurableMul₂ M] (μ : Measure M) (ν : Measure M) (ρ : Measure M) [SFinite μ]
    [SFinite ν] [SFinite ρ] : μ ∗ (ν + ρ) = μ ∗ ν + μ ∗ ρ 
Arguments: {M : Type u_1} [Monoid M] [MeasurableSpace M] [MeasurableMul₂ M] (μ ν ρ : MeasureTheory.Measure M) [MeasureTheory.SFinite μ] [MeasureTheory.SFinite ν] [MeasureTheory.SFinite ρ]
Goal: μ.mconv (ν + ρ) = μ.mconv ν + μ.mconv ρ
-- MeasureTheory.Measure.conv_add
@[to_additive conv_add]
theorem mconv_add [MeasurableMul₂ M] (μ : Measure M) (ν : Measure M) (ρ : Measure M) [SFinite μ]
    [SFinite ν] [SFinite ρ] : μ ∗ (ν + ρ) = μ ∗ ν + μ ∗ ρ 
Arguments: {M : Type u_1} [AddMonoid M] [MeasurableSpace M] [MeasurableAdd₂ M] (μ ν ρ : MeasureTheory.Measure M) [MeasureTheory.SFinite μ] [MeasureTheory.SFinite ν] [MeasureTheory.SFinite ρ]
Goal: μ.conv (ν + ρ) = μ.conv ν + μ.conv ρ
-- MeasureTheory.Measure.add_conv
@[to_additive add_conv]
theorem add_mconv [MeasurableMul₂ M] (μ : Measure M) (ν : Measure M) (ρ : Measure M) [SFinite μ]
    [SFinite ν] [SFinite ρ] : (μ + ν) ∗ ρ = μ ∗ ρ + ν ∗ ρ 
Arguments: {M : Type u_1} [AddMonoid M] [MeasurableSpace M] [MeasurableAdd₂ M] (μ ν ρ : MeasureTheory.Measure M) [MeasureTheory.SFinite μ] [MeasureTheory.SFinite ν] [MeasureTheory.SFinite ρ]
Goal: (μ + ν).conv ρ = μ.conv ρ + ν.conv ρ
-- MeasureTheory.Measure.add_mconv
@[to_additive add_conv]
theorem add_mconv [MeasurableMul₂ M] (μ : Measure M) (ν : Measure M) (ρ : Measure M) [SFinite μ]
    [SFinite ν] [SFinite ρ] : (μ + ν) ∗ ρ = μ ∗ ρ + ν ∗ ρ 
Arguments: {M : Type u_1} [Monoid M] [MeasurableSpace M] [MeasurableMul₂ M] (μ ν ρ : MeasureTheory.Measure M) [MeasureTheory.SFinite μ] [MeasureTheory.SFinite ν] [MeasureTheory.SFinite ρ]
Goal: (μ + ν).mconv ρ = μ.mconv ρ + ν.mconv ρ
-- MeasureTheory.Measure.mconv_comm
@[to_additive conv_comm]
theorem mconv_comm {M : Type*} [CommMonoid M] [MeasurableSpace M] [MeasurableMul₂ M] (μ : Measure M)
    (ν : Measure M) [SFinite μ] [SFinite ν] : μ ∗ ν = ν ∗ μ 
Arguments: {M : Type u_2} [CommMonoid M] [MeasurableSpace M] [MeasurableMul₂ M] (μ ν : MeasureTheory.Measure M) [MeasureTheory.SFinite μ] [MeasureTheory.SFinite ν]
Goal: μ.mconv ν = ν.mconv μ
-- MeasureTheory.Measure.conv_comm
@[to_additive conv_comm]
theorem mconv_comm {M : Type*} [CommMonoid M] [MeasurableSpace M] [MeasurableMul₂ M] (μ : Measure M)
    (ν : Measure M) [SFinite μ] [SFinite ν] : μ ∗ ν = ν ∗ μ 
Arguments: {M : Type u_2} [AddCommMonoid M] [MeasurableSpace M] [MeasurableAdd₂ M] (μ ν : MeasureTheory.Measure M) [MeasureTheory.SFinite μ] [MeasureTheory.SFinite ν]
Goal: μ.conv ν = ν.conv μ
-- MeasureTheory.Measure.InnerRegularWRT.measure_eq_iSup
theorem measure_eq_iSup (H : InnerRegularWRT μ p q) (hU : q U) :
    μ U = ⨆ (K) (_ : K ⊆ U) (_ : p K), μ K 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p q : Set α → Prop} {U : Set α} (H : μ.InnerRegularWRT p q) (hU : q U)
Goal: μ U = ⨆ K, ⨆ (_ : K ⊆ U), ⨆ (_ : p K), μ K
-- MeasureTheory.Measure.InnerRegularWRT.exists_subset_lt_add
theorem exists_subset_lt_add (H : InnerRegularWRT μ p q) (h0 : p ∅) (hU : q U) (hμU : μ U ≠ ∞)
    (hε : ε ≠ 0) : ∃ K, K ⊆ U ∧ p K ∧ μ U < μ K + ε 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p q : Set α → Prop} {U : Set α} {ε : ℝ≥0∞} (H : μ.InnerRegularWRT p q) (h0 : p ∅) (hU : q U) (hμU : μ U ≠ ⊤) (hε : ε ≠ 0)
Goal: ∃ K ⊆ U, p K ∧ μ U < μ K + ε
-- MeasureTheory.Measure.InnerRegularWRT.map
protected theorem map {α β} [MeasurableSpace α] [MeasurableSpace β]
    {μ : Measure α} {pa qa : Set α → Prop}
    (H : InnerRegularWRT μ pa qa) {f : α → β} (hf : AEMeasurable f μ) {pb qb : Set β → Prop}
    (hAB : ∀ U, qb U → qa (f ⁻¹' U)) (hAB' : ∀ K, pa K → pb (f '' K))
    (hB₂ : ∀ U, qb U → MeasurableSet U) :
    InnerRegularWRT (map f μ) pb qb 
Arguments: {α : Type u_2} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {pa qa : Set α → Prop} (H : μ.InnerRegularWRT pa qa) {f : α → β} (hf : AEMeasurable f μ) {pb qb : Set β → Prop} (hAB : ∀ (U : Set β), qb U → qa (f ⁻¹' U)) (hAB' : ∀ (K : Set α), pa K → pb (f '' K)) (hB₂ : ∀ (U : Set β), qb U → MeasurableSet U)
Goal: (MeasureTheory.Measure.map f μ).InnerRegularWRT pb qb
-- MeasureTheory.Measure.InnerRegularWRT.map'
theorem map' {α β} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α} {pa qa : Set α → Prop}
    (H : InnerRegularWRT μ pa qa) (f : α ≃ᵐ β) {pb qb : Set β → Prop}
    (hAB : ∀ U, qb U → qa (f ⁻¹' U)) (hAB' : ∀ K, pa K → pb (f '' K)) :
    InnerRegularWRT (map f μ) pb qb 
Arguments: {α : Type u_2} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {pa qa : Set α → Prop} (H : μ.InnerRegularWRT pa qa) (f : α ≃ᵐ β) {pb qb : Set β → Prop} (hAB : ∀ (U : Set β), qb U → qa (⇑f ⁻¹' U)) (hAB' : ∀ (K : Set α), pa K → pb (⇑f '' K))
Goal: (MeasureTheory.Measure.map (⇑f) μ).InnerRegularWRT pb qb
-- MeasureTheory.Measure.InnerRegularWRT.smul
theorem smul (H : InnerRegularWRT μ p q) (c : ℝ≥0∞) : InnerRegularWRT (c • μ) p q 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p q : Set α → Prop} (H : μ.InnerRegularWRT p q) (c : ℝ≥0∞)
Goal: (c • μ).InnerRegularWRT p q
-- MeasureTheory.Measure.InnerRegularWRT.trans
theorem trans {q' : Set α → Prop} (H : InnerRegularWRT μ p q) (H' : InnerRegularWRT μ q q') :
    InnerRegularWRT μ p q' 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p q q' : Set α → Prop} (H : μ.InnerRegularWRT p q) (H' : μ.InnerRegularWRT q q')
Goal: μ.InnerRegularWRT p q'
-- MeasureTheory.Measure.InnerRegularWRT.rfl
theorem rfl {p : Set α → Prop} : InnerRegularWRT μ p p 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : Set α → Prop}
Goal: μ.InnerRegularWRT p p
-- MeasureTheory.Measure.InnerRegularWRT.of_imp
theorem of_imp (h : ∀ s, q s → p s) : InnerRegularWRT μ p q 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p q : Set α → Prop} (h : ∀ (s : Set α), q s → p s)
Goal: μ.InnerRegularWRT p q
-- MeasureTheory.Measure.InnerRegularWRT.mono
theorem mono {p' q' : Set α → Prop} (H : InnerRegularWRT μ p q)
    (h : ∀ s, q' s → q s) (h' : ∀ s, p s → p' s) : InnerRegularWRT μ p' q' 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p q p' q' : Set α → Prop} (H : μ.InnerRegularWRT p q) (h : ∀ (s : Set α), q' s → q s) (h' : ∀ (s : Set α), p s → p' s)
Goal: μ.InnerRegularWRT p' q'
-- MeasureTheory.Measure.OuterRegular.outerRegular
class OuterRegular (μ : Measure α) : Prop where
  protected outerRegular :
    ∀ ⦃A : Set α⦄, MeasurableSet A → ∀ r > μ A, ∃ U, U ⊇ A ∧ IsOpen U ∧ μ U < r
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [self : μ.OuterRegular] ⦃A : Set α⦄
Goal: MeasurableSet A → ∀ r > μ A, ∃ U ⊇ A, IsOpen U ∧ μ U < r
-- MeasureTheory.Measure.Regular.innerRegular
class Regular (μ : Measure α) extends IsFiniteMeasureOnCompacts μ, OuterRegular μ : Prop where
  innerRegular : InnerRegularWRT μ IsCompact IsOpen
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [self : μ.Regular]
Goal: μ.InnerRegularWRT IsCompact IsOpen
-- MeasureTheory.Measure.WeaklyRegular.innerRegular
class WeaklyRegular (μ : Measure α) extends OuterRegular μ : Prop where
  protected innerRegular : InnerRegularWRT μ IsClosed IsOpen
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [self : μ.WeaklyRegular]
Goal: μ.InnerRegularWRT IsClosed IsOpen
-- MeasureTheory.Measure.InnerRegular.innerRegular
class InnerRegular (μ : Measure α) : Prop where
  protected innerRegular : InnerRegularWRT μ IsCompact (fun s ↦ MeasurableSet s)
Arguments: {α : Type u_1} [MeasurableSpace α] [TopologicalSpace α] {μ : MeasureTheory.Measure α} [self : μ.InnerRegular]
Goal: μ.InnerRegularWRT IsCompact fun s => MeasurableSet s