ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Measure.smul_measure_isMulInvariant_le_of_isCompact_closure
@[to_additive smul_measure_isAddInvariant_le_of_isCompact_closure
"If an invariant measure is inner regular, then it gives less mass to sets with compact closure
than any other invariant measure, up to the scalar `addHaarScalarFactor μ' μ`.

Auxiliary lemma in the proof of the more general
`measure_isAddInvariant_eq_smul_of_isCompact_closure`, which gives equality for any
set with compact closure."]
lemma smul_measure_isMulInvariant_le_of_isCompact_closure [LocallyCompactSpace G]
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ']
    [InnerRegularCompactLTTop μ]
    {s : Set G} (hs : MeasurableSet s) (h's : IsCompact (closure s)) :
    haarScalarFactor μ' μ • μ s ≤ μ' s 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsMulLeftInvariant] [μ.InnerRegularCompactLTTop] {s : Set G} (hs : MeasurableSet s) (h's : IsCompact (closure s))
Goal: μ'.haarScalarFactor μ • μ s ≤ μ' s
-- MeasureTheory.Measure.smul_measure_isAddInvariant_le_of_isCompact_closure
@[to_additive smul_measure_isAddInvariant_le_of_isCompact_closure
"If an invariant measure is inner regular, then it gives less mass to sets with compact closure
than any other invariant measure, up to the scalar `addHaarScalarFactor μ' μ`.

Auxiliary lemma in the proof of the more general
`measure_isAddInvariant_eq_smul_of_isCompact_closure`, which gives equality for any
set with compact closure."]
lemma smul_measure_isMulInvariant_le_of_isCompact_closure [LocallyCompactSpace G]
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ']
    [InnerRegularCompactLTTop μ]
    {s : Set G} (hs : MeasurableSet s) (h's : IsCompact (closure s)) :
    haarScalarFactor μ' μ • μ s ≤ μ' s 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsAddLeftInvariant] [μ.InnerRegularCompactLTTop] {s : Set G} (hs : MeasurableSet s) (h's : IsCompact (closure s))
Goal: μ'.addHaarScalarFactor μ • μ s ≤ μ' s
-- MeasureTheory.Measure.measure_isAddInvariant_eq_smul_of_isCompact_closure_of_innerRegularCompactLTTop
@[to_additive measure_isAddInvariant_eq_smul_of_isCompact_closure_of_innerRegularCompactLTTop
" If an invariant measure is inner regular, then it gives the same mass to measurable sets with
compact closure as any other invariant measure, up to the scalar `addHaarScalarFactor μ' μ`.

Auxiliary lemma in the proof of the more general
`measure_isAddInvariant_eq_smul_of_isCompact_closure`, which works for any set with
compact closure, and removes the inner regularity assumption."]
lemma measure_isMulInvariant_eq_smul_of_isCompact_closure_of_innerRegularCompactLTTop
    [LocallyCompactSpace G]
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ']
    [InnerRegularCompactLTTop μ]
    {s : Set G} (hs : MeasurableSet s) (h's : IsCompact (closure s)) :
    μ' s = haarScalarFactor μ' μ • μ s 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsAddLeftInvariant] [μ.InnerRegularCompactLTTop] {s : Set G} (hs : MeasurableSet s) (h's : IsCompact (closure s))
Goal: μ' s = μ'.addHaarScalarFactor μ • μ s
-- MeasureTheory.Measure.measure_isMulInvariant_eq_smul_of_isCompact_closure_of_innerRegularCompactLTTop
@[to_additive measure_isAddInvariant_eq_smul_of_isCompact_closure_of_innerRegularCompactLTTop
" If an invariant measure is inner regular, then it gives the same mass to measurable sets with
compact closure as any other invariant measure, up to the scalar `addHaarScalarFactor μ' μ`.

Auxiliary lemma in the proof of the more general
`measure_isAddInvariant_eq_smul_of_isCompact_closure`, which works for any set with
compact closure, and removes the inner regularity assumption."]
lemma measure_isMulInvariant_eq_smul_of_isCompact_closure_of_innerRegularCompactLTTop
    [LocallyCompactSpace G]
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ']
    [InnerRegularCompactLTTop μ]
    {s : Set G} (hs : MeasurableSet s) (h's : IsCompact (closure s)) :
    μ' s = haarScalarFactor μ' μ • μ s 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsMulLeftInvariant] [μ.InnerRegularCompactLTTop] {s : Set G} (hs : MeasurableSet s) (h's : IsCompact (closure s))
Goal: μ' s = μ'.haarScalarFactor μ • μ s
-- MeasureTheory.Measure.measure_isAddInvariant_eq_smul_of_isCompact_closure_of_measurableSet
@[to_additive measure_isAddInvariant_eq_smul_of_isCompact_closure_of_measurableSet
"Given an invariant measure then it gives the same mass to measurable sets with
compact closure as any other invariant measure, up to the scalar `addHaarScalarFactor μ' μ`.

Auxiliary lemma in the proof of the more general
`measure_isAddInvariant_eq_smul_of_isCompact_closure`, which removes the
measurability assumption."]
lemma measure_isMulInvariant_eq_smul_of_isCompact_closure_of_measurableSet [LocallyCompactSpace G]
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ']
    {s : Set G} (hs : MeasurableSet s) (h's : IsCompact (closure s)) :
    μ' s = haarScalarFactor μ' μ • μ s 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsAddLeftInvariant] {s : Set G} (hs : MeasurableSet s) (h's : IsCompact (closure s))
Goal: μ' s = μ'.addHaarScalarFactor μ • μ s
-- MeasureTheory.Measure.measure_isMulInvariant_eq_smul_of_isCompact_closure_of_measurableSet
@[to_additive measure_isAddInvariant_eq_smul_of_isCompact_closure_of_measurableSet
"Given an invariant measure then it gives the same mass to measurable sets with
compact closure as any other invariant measure, up to the scalar `addHaarScalarFactor μ' μ`.

Auxiliary lemma in the proof of the more general
`measure_isAddInvariant_eq_smul_of_isCompact_closure`, which removes the
measurability assumption."]
lemma measure_isMulInvariant_eq_smul_of_isCompact_closure_of_measurableSet [LocallyCompactSpace G]
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ']
    {s : Set G} (hs : MeasurableSet s) (h's : IsCompact (closure s)) :
    μ' s = haarScalarFactor μ' μ • μ s 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsMulLeftInvariant] {s : Set G} (hs : MeasurableSet s) (h's : IsCompact (closure s))
Goal: μ' s = μ'.haarScalarFactor μ • μ s
-- MeasureTheory.Measure.measure_isMulInvariant_eq_smul_of_isCompact_closure
@[to_additive measure_isAddInvariant_eq_smul_of_isCompact_closure
"**Uniqueness of left-invariant measures**:
Given two left-invariant measures which are finite on compacts, they coincide in the following
sense: they give the same value to sets with compact closure, up to the multiplicative
constant `addHaarScalarFactor μ' μ`. "]
theorem measure_isMulInvariant_eq_smul_of_isCompact_closure [LocallyCompactSpace G]
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ']
    {s : Set G} (h's : IsCompact (closure s)) :
    μ' s = haarScalarFactor μ' μ • μ s 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsMulLeftInvariant] {s : Set G} (h's : IsCompact (closure s))
Goal: μ' s = μ'.haarScalarFactor μ • μ s
-- MeasureTheory.Measure.measure_isAddInvariant_eq_smul_of_isCompact_closure
@[to_additive measure_isAddInvariant_eq_smul_of_isCompact_closure
"**Uniqueness of left-invariant measures**:
Given two left-invariant measures which are finite on compacts, they coincide in the following
sense: they give the same value to sets with compact closure, up to the multiplicative
constant `addHaarScalarFactor μ' μ`. "]
theorem measure_isMulInvariant_eq_smul_of_isCompact_closure [LocallyCompactSpace G]
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ']
    {s : Set G} (h's : IsCompact (closure s)) :
    μ' s = haarScalarFactor μ' μ • μ s 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsAddLeftInvariant] {s : Set G} (h's : IsCompact (closure s))
Goal: μ' s = μ'.addHaarScalarFactor μ • μ s
-- MeasureTheory.Measure.isMulInvariant_eq_smul_of_compactSpace
@[to_additive isAddInvariant_eq_smul_of_compactSpace]
lemma isMulInvariant_eq_smul_of_compactSpace [CompactSpace G] (μ' μ : Measure G)
    [IsHaarMeasure μ] [IsMulLeftInvariant μ'] [IsFiniteMeasureOnCompacts μ'] :
    μ' = haarScalarFactor μ' μ • μ 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] [CompactSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [μ'.IsMulLeftInvariant] [MeasureTheory.IsFiniteMeasureOnCompacts μ']
Goal: μ' = μ'.haarScalarFactor μ • μ
-- MeasureTheory.Measure.isAddInvariant_eq_smul_of_compactSpace
@[to_additive isAddInvariant_eq_smul_of_compactSpace]
lemma isMulInvariant_eq_smul_of_compactSpace [CompactSpace G] (μ' μ : Measure G)
    [IsHaarMeasure μ] [IsMulLeftInvariant μ'] [IsFiniteMeasureOnCompacts μ'] :
    μ' = haarScalarFactor μ' μ • μ 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] [CompactSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [μ'.IsAddLeftInvariant] [MeasureTheory.IsFiniteMeasureOnCompacts μ']
Goal: μ' = μ'.addHaarScalarFactor μ • μ
-- MeasureTheory.Measure.isHaarMeasure_eq_of_isProbabilityMeasure
@[to_additive]
lemma isHaarMeasure_eq_of_isProbabilityMeasure [LocallyCompactSpace G] (μ' μ : Measure G)
    [IsProbabilityMeasure μ] [IsProbabilityMeasure μ'] [IsHaarMeasure μ] [IsHaarMeasure μ'] :
    μ' = μ 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] [μ.IsHaarMeasure] [μ'.IsHaarMeasure]
Goal: μ' = μ
-- MeasureTheory.Measure.isAddHaarMeasure_eq_of_isProbabilityMeasure
@[to_additive]
lemma isHaarMeasure_eq_of_isProbabilityMeasure [LocallyCompactSpace G] (μ' μ : Measure G)
    [IsProbabilityMeasure μ] [IsProbabilityMeasure μ'] [IsHaarMeasure μ] [IsHaarMeasure μ'] :
    μ' = μ 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] [μ.IsAddHaarMeasure] [μ'.IsAddHaarMeasure]
Goal: μ' = μ
-- MeasureTheory.Measure.measure_isAddHaarMeasure_eq_smul_of_isEverywherePos
@[to_additive measure_isAddHaarMeasure_eq_smul_of_isEverywherePos]
theorem measure_isHaarMeasure_eq_smul_of_isEverywherePos [LocallyCompactSpace G]
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsHaarMeasure μ']
    {s : Set G} (hs : MeasurableSet s) (h's : IsEverywherePos μ s) :
    μ' s = haarScalarFactor μ' μ • μ s 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [μ'.IsAddHaarMeasure] {s : Set G} (hs : MeasurableSet s) (h's : μ.IsEverywherePos s)
Goal: μ' s = μ'.addHaarScalarFactor μ • μ s
-- MeasureTheory.Measure.measure_isHaarMeasure_eq_smul_of_isEverywherePos
@[to_additive measure_isAddHaarMeasure_eq_smul_of_isEverywherePos]
theorem measure_isHaarMeasure_eq_smul_of_isEverywherePos [LocallyCompactSpace G]
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsHaarMeasure μ']
    {s : Set G} (hs : MeasurableSet s) (h's : IsEverywherePos μ s) :
    μ' s = haarScalarFactor μ' μ • μ s 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [μ'.IsHaarMeasure] {s : Set G} (hs : MeasurableSet s) (h's : μ.IsEverywherePos s)
Goal: μ' s = μ'.haarScalarFactor μ • μ s
-- MeasureTheory.Measure.measure_isHaarMeasure_eq_smul_of_isOpen
@[to_additive measure_isAddHaarMeasure_eq_smul_of_isOpen
"**Uniqueness of Haar measures**:
Given two additive Haar measures, they coincide in the following sense: they give the same value to
open sets, up to the multiplicative constant `addHaarScalarFactor μ' μ`."]
theorem measure_isHaarMeasure_eq_smul_of_isOpen [LocallyCompactSpace G]
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsHaarMeasure μ'] {s : Set G} (hs : IsOpen s) :
    μ' s = haarScalarFactor μ' μ • μ s 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [μ'.IsHaarMeasure] {s : Set G} (hs : IsOpen s)
Goal: μ' s = μ'.haarScalarFactor μ • μ s
-- MeasureTheory.Measure.measure_isAddHaarMeasure_eq_smul_of_isOpen
@[to_additive measure_isAddHaarMeasure_eq_smul_of_isOpen
"**Uniqueness of Haar measures**:
Given two additive Haar measures, they coincide in the following sense: they give the same value to
open sets, up to the multiplicative constant `addHaarScalarFactor μ' μ`."]
theorem measure_isHaarMeasure_eq_smul_of_isOpen [LocallyCompactSpace G]
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsHaarMeasure μ'] {s : Set G} (hs : IsOpen s) :
    μ' s = haarScalarFactor μ' μ • μ s 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [μ'.IsAddHaarMeasure] {s : Set G} (hs : IsOpen s)
Goal: μ' s = μ'.addHaarScalarFactor μ • μ s
-- MeasureTheory.Measure.measure_isAddLeftInvariant_eq_vadd_of_ne_top
@[to_additive]
lemma measure_isMulLeftInvariant_eq_smul_of_ne_top [LocallyCompactSpace G]
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ']
    [InnerRegularCompactLTTop μ] [InnerRegularCompactLTTop μ'] {s : Set G}
    (hs : μ s ≠ ∞) (h's : μ' s ≠ ∞) : μ' s = haarScalarFactor μ' μ • μ s 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsAddLeftInvariant] [μ.InnerRegularCompactLTTop] [μ'.InnerRegularCompactLTTop] {s : Set G} (hs : μ s ≠ ⊤) (h's : μ' s ≠ ⊤)
Goal: μ' s = μ'.addHaarScalarFactor μ • μ s
-- MeasureTheory.Measure.measure_isMulLeftInvariant_eq_smul_of_ne_top
@[to_additive]
lemma measure_isMulLeftInvariant_eq_smul_of_ne_top [LocallyCompactSpace G]
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ']
    [InnerRegularCompactLTTop μ] [InnerRegularCompactLTTop μ'] {s : Set G}
    (hs : μ s ≠ ∞) (h's : μ' s ≠ ∞) : μ' s = haarScalarFactor μ' μ • μ s 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsMulLeftInvariant] [μ.InnerRegularCompactLTTop] [μ'.InnerRegularCompactLTTop] {s : Set G} (hs : μ s ≠ ⊤) (h's : μ' s ≠ ⊤)
Goal: μ' s = μ'.haarScalarFactor μ • μ s
-- MeasureTheory.Measure.isMulLeftInvariant_eq_smul_of_innerRegular
@[to_additive isAddLeftInvariant_eq_smul_of_innerRegular]
lemma isMulLeftInvariant_eq_smul_of_innerRegular [LocallyCompactSpace G]
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ']
    [InnerRegular μ] [InnerRegular μ'] :
    μ' = haarScalarFactor μ' μ • μ 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsMulLeftInvariant] [μ.InnerRegular] [μ'.InnerRegular]
Goal: μ' = μ'.haarScalarFactor μ • μ
-- MeasureTheory.Measure.isAddLeftInvariant_eq_smul_of_innerRegular
@[to_additive isAddLeftInvariant_eq_smul_of_innerRegular]
lemma isMulLeftInvariant_eq_smul_of_innerRegular [LocallyCompactSpace G]
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ']
    [InnerRegular μ] [InnerRegular μ'] :
    μ' = haarScalarFactor μ' μ • μ 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsAddLeftInvariant] [μ.InnerRegular] [μ'.InnerRegular]
Goal: μ' = μ'.addHaarScalarFactor μ • μ
-- MeasureTheory.Measure.isAddLeftInvariant_eq_smul_of_regular
@[to_additive isAddLeftInvariant_eq_smul_of_regular]
lemma isMulLeftInvariant_eq_smul_of_regular [LocallyCompactSpace G]
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ']
    [Regular μ] [Regular μ'] :
    μ' = haarScalarFactor μ' μ • μ 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsAddLeftInvariant] [μ.Regular] [μ'.Regular]
Goal: μ' = μ'.addHaarScalarFactor μ • μ
-- MeasureTheory.Measure.isMulLeftInvariant_eq_smul_of_regular
@[to_additive isAddLeftInvariant_eq_smul_of_regular]
lemma isMulLeftInvariant_eq_smul_of_regular [LocallyCompactSpace G]
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ']
    [Regular μ] [Regular μ'] :
    μ' = haarScalarFactor μ' μ • μ 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsMulLeftInvariant] [μ.Regular] [μ'.Regular]
Goal: μ' = μ'.haarScalarFactor μ • μ
-- MeasureTheory.Measure.isMulLeftInvariant_eq_smul
@[to_additive isAddLeftInvariant_eq_smul]
lemma isMulLeftInvariant_eq_smul [LocallyCompactSpace G] [SecondCountableTopology G]
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ'] :
    μ' = haarScalarFactor μ' μ • μ 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] [SecondCountableTopology G] (μ' μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsMulLeftInvariant]
Goal: μ' = μ'.haarScalarFactor μ • μ
-- MeasureTheory.Measure.isAddLeftInvariant_eq_smul
@[to_additive isAddLeftInvariant_eq_smul]
lemma isMulLeftInvariant_eq_smul [LocallyCompactSpace G] [SecondCountableTopology G]
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsFiniteMeasureOnCompacts μ'] [IsMulLeftInvariant μ'] :
    μ' = haarScalarFactor μ' μ • μ 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] [SecondCountableTopology G] (μ' μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [MeasureTheory.IsFiniteMeasureOnCompacts μ'] [μ'.IsAddLeftInvariant]
Goal: μ' = μ'.addHaarScalarFactor μ • μ
-- MeasureTheory.Measure.absolutelyContinuous_isAddHaarMeasure
@[to_additive
"An invariant measure is absolutely continuous with respect to an additive Haar measure. "]
theorem absolutelyContinuous_isHaarMeasure [LocallyCompactSpace G]
    [SecondCountableTopology G] (μ ν : Measure G)
    [SigmaFinite μ] [IsMulLeftInvariant μ] [IsHaarMeasure ν] : μ ≪ ν 
Arguments: {G : Type u_1} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] [SecondCountableTopology G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite μ] [μ.IsAddLeftInvariant] [ν.IsAddHaarMeasure]
Goal: μ ≪ ν
-- MeasureTheory.Measure.absolutelyContinuous_isHaarMeasure
@[to_additive
"An invariant measure is absolutely continuous with respect to an additive Haar measure. "]
theorem absolutelyContinuous_isHaarMeasure [LocallyCompactSpace G]
    [SecondCountableTopology G] (μ ν : Measure G)
    [SigmaFinite μ] [IsMulLeftInvariant μ] [IsHaarMeasure ν] : μ ≪ ν 
Arguments: {G : Type u_1} [TopologicalSpace G] [Group G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] [LocallyCompactSpace G] [SecondCountableTopology G] (μ ν : MeasureTheory.Measure G) [MeasureTheory.SigmaFinite μ] [μ.IsMulLeftInvariant] [ν.IsHaarMeasure]
Goal: μ ≪ ν
-- MeasureTheory.Measure.measurePreserving_zsmul
@[to_additive]
theorem measurePreserving_zpow [CompactSpace G] [RootableBy G ℤ] {n : ℤ} (hn : n ≠ 0) :
    MeasurePreserving (fun g : G => g ^ n) μ μ 
Arguments: {G : Type u_1} [AddCommGroup G] [TopologicalSpace G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] (μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [CompactSpace G] [DivisibleBy G ℤ] {n : ℤ} (hn : n ≠ 0)
Goal: MeasureTheory.MeasurePreserving (fun g => n • g) μ μ
-- MeasureTheory.Measure.measurePreserving_zpow
@[to_additive]
theorem measurePreserving_zpow [CompactSpace G] [RootableBy G ℤ] {n : ℤ} (hn : n ≠ 0) :
    MeasurePreserving (fun g : G => g ^ n) μ μ 
Arguments: {G : Type u_1} [CommGroup G] [TopologicalSpace G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] (μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [CompactSpace G] [RootableBy G ℤ] {n : ℤ} (hn : n ≠ 0)
Goal: MeasureTheory.MeasurePreserving (fun g => g ^ n) μ μ
-- MeasureTheory.Measure.MeasurePreserving.zsmul
@[to_additive]
theorem MeasurePreserving.zpow [CompactSpace G] [RootableBy G ℤ]
    {n : ℤ} (hn : n ≠ 0) {X : Type*}
    [MeasurableSpace X] {μ' : Measure X} {f : X → G} (hf : MeasurePreserving f μ' μ) :
    MeasurePreserving (fun x => f x ^ n) μ' μ 
Arguments: {G : Type u_1} [AddCommGroup G] [TopologicalSpace G] [TopologicalAddGroup G] [MeasurableSpace G] [BorelSpace G] (μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [CompactSpace G] [DivisibleBy G ℤ] {n : ℤ} (hn : n ≠ 0) {X : Type u_2} [MeasurableSpace X] {μ' : MeasureTheory.Measure X} {f : X → G} (hf : MeasureTheory.MeasurePreserving f μ' μ)
Goal: MeasureTheory.MeasurePreserving (fun x => n • f x) μ' μ
-- MeasureTheory.Measure.MeasurePreserving.zpow
@[to_additive]
theorem MeasurePreserving.zpow [CompactSpace G] [RootableBy G ℤ]
    {n : ℤ} (hn : n ≠ 0) {X : Type*}
    [MeasurableSpace X] {μ' : Measure X} {f : X → G} (hf : MeasurePreserving f μ' μ) :
    MeasurePreserving (fun x => f x ^ n) μ' μ 
Arguments: {G : Type u_1} [CommGroup G] [TopologicalSpace G] [TopologicalGroup G] [MeasurableSpace G] [BorelSpace G] (μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [CompactSpace G] [RootableBy G ℤ] {n : ℤ} (hn : n ≠ 0) {X : Type u_2} [MeasurableSpace X] {μ' : MeasureTheory.Measure X} {f : X → G} (hf : MeasureTheory.MeasurePreserving f μ' μ)
Goal: MeasureTheory.MeasurePreserving (fun x => f x ^ n) μ' μ