l3lab/ntp-mathlib huggingface dataset:
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite
on compacts and inner regular, they coincide up to a multiplicative constant. -/
@[to_additive isAddLeftInvariant_eq_smul_of_innerRegular]
lemma isMulLeftInvariant_eq_smul_of_innerRegular
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [InnerRegular μ] [InnerRegular μ'] :
    ∃ (c : ℝ≥0), μ' = c • μ := by
  rcases measure_isMulLeftInvariant_eq_smul_of_ne_top μ' μ with ⟨c, hc⟩
  refine ⟨c, ?_⟩
  

Proof state:
case intro
G : Type u_1
inst✝¹² : TopologicalSpace G
inst✝¹¹ : LocallyCompactSpace G
inst✝¹⁰ : Group G
inst✝⁹ : TopologicalGroup G
inst✝⁸ : MeasurableSpace G
inst✝⁷ : BorelSpace G
μ' μ : Measure G
inst✝⁶ : IsFiniteMeasureOnCompacts μ
inst✝⁵ : IsFiniteMeasureOnCompacts μ'
inst✝⁴ : IsMulLeftInvariant μ
inst✝³ : IsMulLeftInvariant μ'
inst✝² : IsOpenPosMeasure μ
inst✝¹ : InnerRegular μ
inst✝ : InnerRegular μ'
c : ℝ≥0
hc : ∀ (s : Set G), ↑↑μ s < ⊤ → ↑↑μ' s < ⊤ → ↑↑μ' s = ↑↑(c • μ) s
⊢ μ' = c • μ

Next tactic:
ext s hs
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite
on compacts and inner regular, they coincide up to a multiplicative constant. -/
@[to_additive isAddLeftInvariant_eq_smul_of_innerRegular]
lemma isMulLeftInvariant_eq_smul_of_innerRegular
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [InnerRegular μ] [InnerRegular μ'] :
    ∃ (c : ℝ≥0), μ' = c • μ := by
  rcases measure_isMulLeftInvariant_eq_smul_of_ne_top μ' μ with ⟨c, hc⟩
  refine ⟨c, ?_⟩
  ext s hs
  

Proof state:
case intro.h
G : Type u_1
inst✝¹² : TopologicalSpace G
inst✝¹¹ : LocallyCompactSpace G
inst✝¹⁰ : Group G
inst✝⁹ : TopologicalGroup G
inst✝⁸ : MeasurableSpace G
inst✝⁷ : BorelSpace G
μ' μ : Measure G
inst✝⁶ : IsFiniteMeasureOnCompacts μ
inst✝⁵ : IsFiniteMeasureOnCompacts μ'
inst✝⁴ : IsMulLeftInvariant μ
inst✝³ : IsMulLeftInvariant μ'
inst✝² : IsOpenPosMeasure μ
inst✝¹ : InnerRegular μ
inst✝ : InnerRegular μ'
c : ℝ≥0
hc : ∀ (s : Set G), ↑↑μ s < ⊤ → ↑↑μ' s < ⊤ → ↑↑μ' s = ↑↑(c • μ) s
s : Set G
hs : MeasurableSet s
⊢ ↑↑μ' s = ↑↑(c • μ) s

Next tactic:
rw [hs.measure_eq_iSup_isCompact, hs.measure_eq_iSup_isCompact]
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite
on compacts and inner regular, they coincide up to a multiplicative constant. -/
@[to_additive isAddLeftInvariant_eq_smul_of_innerRegular]
lemma isMulLeftInvariant_eq_smul_of_innerRegular
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [InnerRegular μ] [InnerRegular μ'] :
    ∃ (c : ℝ≥0), μ' = c • μ := by
  rcases measure_isMulLeftInvariant_eq_smul_of_ne_top μ' μ with ⟨c, hc⟩
  refine ⟨c, ?_⟩
  ext s hs
  rw [hs.measure_eq_iSup_isCompact, hs.measure_eq_iSup_isCompact]
  

Proof state:
case intro.h
G : Type u_1
inst✝¹² : TopologicalSpace G
inst✝¹¹ : LocallyCompactSpace G
inst✝¹⁰ : Group G
inst✝⁹ : TopologicalGroup G
inst✝⁸ : MeasurableSpace G
inst✝⁷ : BorelSpace G
μ' μ : Measure G
inst✝⁶ : IsFiniteMeasureOnCompacts μ
inst✝⁵ : IsFiniteMeasureOnCompacts μ'
inst✝⁴ : IsMulLeftInvariant μ
inst✝³ : IsMulLeftInvariant μ'
inst✝² : IsOpenPosMeasure μ
inst✝¹ : InnerRegular μ
inst✝ : InnerRegular μ'
c : ℝ≥0
hc : ∀ (s : Set G), ↑↑μ s < ⊤ → ↑↑μ' s < ⊤ → ↑↑μ' s = ↑↑(c • μ) s
s : Set G
hs : MeasurableSet s
⊢ ⨆ K, ⨆ (_ : K ⊆ s), ⨆ (_ : IsCompact K), ↑↑μ' K = ⨆ K, ⨆ (_ : K ⊆ s), ⨆ (_ : IsCompact K), ↑↑(c • μ) K

Next tactic:
congr! 4 with K _Ks K_comp
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite
on compacts and inner regular, they coincide up to a multiplicative constant. -/
@[to_additive isAddLeftInvariant_eq_smul_of_innerRegular]
lemma isMulLeftInvariant_eq_smul_of_innerRegular
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [InnerRegular μ] [InnerRegular μ'] :
    ∃ (c : ℝ≥0), μ' = c • μ := by
  rcases measure_isMulLeftInvariant_eq_smul_of_ne_top μ' μ with ⟨c, hc⟩
  refine ⟨c, ?_⟩
  ext s hs
  rw [hs.measure_eq_iSup_isCompact, hs.measure_eq_iSup_isCompact]
  congr! 4 with K _Ks K_comp
  

Proof state:
case intro.h.h.e'_4.h.f.f
G : Type u_1
inst✝¹² : TopologicalSpace G
inst✝¹¹ : LocallyCompactSpace G
inst✝¹⁰ : Group G
inst✝⁹ : TopologicalGroup G
inst✝⁸ : MeasurableSpace G
inst✝⁷ : BorelSpace G
μ' μ : Measure G
inst✝⁶ : IsFiniteMeasureOnCompacts μ
inst✝⁵ : IsFiniteMeasureOnCompacts μ'
inst✝⁴ : IsMulLeftInvariant μ
inst✝³ : IsMulLeftInvariant μ'
inst✝² : IsOpenPosMeasure μ
inst✝¹ : InnerRegular μ
inst✝ : InnerRegular μ'
c : ℝ≥0
hc : ∀ (s : Set G), ↑↑μ s < ⊤ → ↑↑μ' s < ⊤ → ↑↑μ' s = ↑↑(c • μ) s
s : Set G
hs : MeasurableSet s
K : Set G
_Ks : K ⊆ s
K_comp : IsCompact K
⊢ ↑↑μ' K = ↑↑(c • μ) K

Next tactic:
exact hc K K_comp.measure_lt_top K_comp.measure_lt_top
/-- **Uniqueness of left-invariant measures**: Two inner regular Haar measures coincide up to a
multiplicative constant. -/
@[to_additive isAddHaarMeasure_eq_smul_of_innerRegular]
lemma isHaarMeasure_eq_smul_of_innerRegular
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsHaarMeasure μ'] [InnerRegular μ] [InnerRegular μ'] :
    ∃ c : ℝ≥0∞, c ≠ 0 ∧ c ≠ ∞ ∧ μ' = c • μ := by
  

Proof state:
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : LocallyCompactSpace G
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
μ' μ : Measure G
inst✝³ : IsHaarMeasure μ
inst✝² : IsHaarMeasure μ'
inst✝¹ : InnerRegular μ
inst✝ : InnerRegular μ'
⊢ ∃ c, c ≠ 0 ∧ c ≠ ⊤ ∧ μ' = c • μ

Next tactic:
rcases isMulLeftInvariant_eq_smul_of_innerRegular μ' μ with ⟨c, hc⟩
/-- **Uniqueness of left-invariant measures**: Two inner regular Haar measures coincide up to a
multiplicative constant. -/
@[to_additive isAddHaarMeasure_eq_smul_of_innerRegular]
lemma isHaarMeasure_eq_smul_of_innerRegular
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsHaarMeasure μ'] [InnerRegular μ] [InnerRegular μ'] :
    ∃ c : ℝ≥0∞, c ≠ 0 ∧ c ≠ ∞ ∧ μ' = c • μ := by
  rcases isMulLeftInvariant_eq_smul_of_innerRegular μ' μ with ⟨c, hc⟩
  

Proof state:
case intro
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : LocallyCompactSpace G
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
μ' μ : Measure G
inst✝³ : IsHaarMeasure μ
inst✝² : IsHaarMeasure μ'
inst✝¹ : InnerRegular μ
inst✝ : InnerRegular μ'
c : ℝ≥0
hc : μ' = c • μ
⊢ ∃ c, c ≠ 0 ∧ c ≠ ⊤ ∧ μ' = c • μ

Next tactic:
refine ⟨c, ?_, ENNReal.coe_ne_top, hc⟩
/-- **Uniqueness of left-invariant measures**: Two inner regular Haar measures coincide up to a
multiplicative constant. -/
@[to_additive isAddHaarMeasure_eq_smul_of_innerRegular]
lemma isHaarMeasure_eq_smul_of_innerRegular
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsHaarMeasure μ'] [InnerRegular μ] [InnerRegular μ'] :
    ∃ c : ℝ≥0∞, c ≠ 0 ∧ c ≠ ∞ ∧ μ' = c • μ := by
  rcases isMulLeftInvariant_eq_smul_of_innerRegular μ' μ with ⟨c, hc⟩
  refine ⟨c, ?_, ENNReal.coe_ne_top, hc⟩
  

Proof state:
case intro
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : LocallyCompactSpace G
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
μ' μ : Measure G
inst✝³ : IsHaarMeasure μ
inst✝² : IsHaarMeasure μ'
inst✝¹ : InnerRegular μ
inst✝ : InnerRegular μ'
c : ℝ≥0
hc : μ' = c • μ
⊢ ↑c ≠ 0

Next tactic:
intro h
/-- **Uniqueness of left-invariant measures**: Two inner regular Haar measures coincide up to a
multiplicative constant. -/
@[to_additive isAddHaarMeasure_eq_smul_of_innerRegular]
lemma isHaarMeasure_eq_smul_of_innerRegular
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsHaarMeasure μ'] [InnerRegular μ] [InnerRegular μ'] :
    ∃ c : ℝ≥0∞, c ≠ 0 ∧ c ≠ ∞ ∧ μ' = c • μ := by
  rcases isMulLeftInvariant_eq_smul_of_innerRegular μ' μ with ⟨c, hc⟩
  refine ⟨c, ?_, ENNReal.coe_ne_top, hc⟩
  intro h
  

Proof state:
case intro
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : LocallyCompactSpace G
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
μ' μ : Measure G
inst✝³ : IsHaarMeasure μ
inst✝² : IsHaarMeasure μ'
inst✝¹ : InnerRegular μ
inst✝ : InnerRegular μ'
c : ℝ≥0
hc : μ' = c • μ
h : ↑c = 0
⊢ False

Next tactic:
simp only [ENNReal.coe_eq_zero.1 h, zero_smul] at hc
/-- **Uniqueness of left-invariant measures**: Two inner regular Haar measures coincide up to a
multiplicative constant. -/
@[to_additive isAddHaarMeasure_eq_smul_of_innerRegular]
lemma isHaarMeasure_eq_smul_of_innerRegular
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsHaarMeasure μ'] [InnerRegular μ] [InnerRegular μ'] :
    ∃ c : ℝ≥0∞, c ≠ 0 ∧ c ≠ ∞ ∧ μ' = c • μ := by
  rcases isMulLeftInvariant_eq_smul_of_innerRegular μ' μ with ⟨c, hc⟩
  refine ⟨c, ?_, ENNReal.coe_ne_top, hc⟩
  intro h
  simp only [ENNReal.coe_eq_zero.1 h, zero_smul] at hc
  

Proof state:
case intro
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : LocallyCompactSpace G
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
μ' μ : Measure G
inst✝³ : IsHaarMeasure μ
inst✝² : IsHaarMeasure μ'
inst✝¹ : InnerRegular μ
inst✝ : InnerRegular μ'
c : ℝ≥0
h : ↑c = 0
hc : μ' = 0
⊢ False

Next tactic:
have : 0 < μ' univ := NeZero.pos (μ' univ)
/-- **Uniqueness of left-invariant measures**: Two inner regular Haar measures coincide up to a
multiplicative constant. -/
@[to_additive isAddHaarMeasure_eq_smul_of_innerRegular]
lemma isHaarMeasure_eq_smul_of_innerRegular
    (μ' μ : Measure G) [IsHaarMeasure μ] [IsHaarMeasure μ'] [InnerRegular μ] [InnerRegular μ'] :
    ∃ c : ℝ≥0∞, c ≠ 0 ∧ c ≠ ∞ ∧ μ' = c • μ := by
  rcases isMulLeftInvariant_eq_smul_of_innerRegular μ' μ with ⟨c, hc⟩
  refine ⟨c, ?_, ENNReal.coe_ne_top, hc⟩
  intro h
  simp only [ENNReal.coe_eq_zero.1 h, zero_smul] at hc
  have : 0 < μ' univ := NeZero.pos (μ' univ)
  

Proof state:
case intro
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : LocallyCompactSpace G
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
μ' μ : Measure G
inst✝³ : IsHaarMeasure μ
inst✝² : IsHaarMeasure μ'
inst✝¹ : InnerRegular μ
inst✝ : InnerRegular μ'
c : ℝ≥0
h : ↑c = 0
hc : μ' = 0
this : 0 < ↑↑μ' univ
⊢ False

Next tactic:
simp [hc] at this
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite
on compacts and regular, they coincide up to a multiplicative constant. -/
@[to_additive isAddLeftInvariant_eq_smul_of_regular]
lemma isMulLeftInvariant_eq_smul_of_regular
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [Regular μ] [Regular μ'] :
    ∃ (c : ℝ≥0), μ' = c • μ := by
  

Proof state:
G : Type u_1
inst✝¹² : TopologicalSpace G
inst✝¹¹ : LocallyCompactSpace G
inst✝¹⁰ : Group G
inst✝⁹ : TopologicalGroup G
inst✝⁸ : MeasurableSpace G
inst✝⁷ : BorelSpace G
μ' μ : Measure G
inst✝⁶ : IsFiniteMeasureOnCompacts μ
inst✝⁵ : IsFiniteMeasureOnCompacts μ'
inst✝⁴ : IsMulLeftInvariant μ
inst✝³ : IsMulLeftInvariant μ'
inst✝² : IsOpenPosMeasure μ
inst✝¹ : Regular μ
inst✝ : Regular μ'
⊢ ∃ c, μ' = c • μ

Next tactic:
rcases measure_isMulLeftInvariant_eq_smul_of_ne_top μ' μ with ⟨c, hc⟩
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite
on compacts and regular, they coincide up to a multiplicative constant. -/
@[to_additive isAddLeftInvariant_eq_smul_of_regular]
lemma isMulLeftInvariant_eq_smul_of_regular
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [Regular μ] [Regular μ'] :
    ∃ (c : ℝ≥0), μ' = c • μ := by
  rcases measure_isMulLeftInvariant_eq_smul_of_ne_top μ' μ with ⟨c, hc⟩
  

Proof state:
case intro
G : Type u_1
inst✝¹² : TopologicalSpace G
inst✝¹¹ : LocallyCompactSpace G
inst✝¹⁰ : Group G
inst✝⁹ : TopologicalGroup G
inst✝⁸ : MeasurableSpace G
inst✝⁷ : BorelSpace G
μ' μ : Measure G
inst✝⁶ : IsFiniteMeasureOnCompacts μ
inst✝⁵ : IsFiniteMeasureOnCompacts μ'
inst✝⁴ : IsMulLeftInvariant μ
inst✝³ : IsMulLeftInvariant μ'
inst✝² : IsOpenPosMeasure μ
inst✝¹ : Regular μ
inst✝ : Regular μ'
c : ℝ≥0
hc : ∀ (s : Set G), ↑↑μ s < ⊤ → ↑↑μ' s < ⊤ → ↑↑μ' s = ↑↑(c • μ) s
⊢ ∃ c, μ' = c • μ

Next tactic:
refine ⟨c, ?_⟩
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite
on compacts and regular, they coincide up to a multiplicative constant. -/
@[to_additive isAddLeftInvariant_eq_smul_of_regular]
lemma isMulLeftInvariant_eq_smul_of_regular
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [Regular μ] [Regular μ'] :
    ∃ (c : ℝ≥0), μ' = c • μ := by
  rcases measure_isMulLeftInvariant_eq_smul_of_ne_top μ' μ with ⟨c, hc⟩
  refine ⟨c, ?_⟩
  

Proof state:
case intro
G : Type u_1
inst✝¹² : TopologicalSpace G
inst✝¹¹ : LocallyCompactSpace G
inst✝¹⁰ : Group G
inst✝⁹ : TopologicalGroup G
inst✝⁸ : MeasurableSpace G
inst✝⁷ : BorelSpace G
μ' μ : Measure G
inst✝⁶ : IsFiniteMeasureOnCompacts μ
inst✝⁵ : IsFiniteMeasureOnCompacts μ'
inst✝⁴ : IsMulLeftInvariant μ
inst✝³ : IsMulLeftInvariant μ'
inst✝² : IsOpenPosMeasure μ
inst✝¹ : Regular μ
inst✝ : Regular μ'
c : ℝ≥0
hc : ∀ (s : Set G), ↑↑μ s < ⊤ → ↑↑μ' s < ⊤ → ↑↑μ' s = ↑↑(c • μ) s
⊢ μ' = c • μ

Next tactic:
have A : ∀ U, IsOpen U → μ' U = (c • μ) U := by
    intro U hU
    rw [hU.measure_eq_iSup_isCompact, hU.measure_eq_iSup_isCompact]
    congr! 4 with K _KU K_comp
    exact hc K K_comp.measure_lt_top K_comp.measure_lt_top
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite
on compacts and regular, they coincide up to a multiplicative constant. -/
@[to_additive isAddLeftInvariant_eq_smul_of_regular]
lemma isMulLeftInvariant_eq_smul_of_regular
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [Regular μ] [Regular μ'] :
    ∃ (c : ℝ≥0), μ' = c • μ := by
  rcases measure_isMulLeftInvariant_eq_smul_of_ne_top μ' μ with ⟨c, hc⟩
  refine ⟨c, ?_⟩
  have A : ∀ U, IsOpen U → μ' U = (c • μ) U := by
    

Proof state:
G : Type u_1
inst✝¹² : TopologicalSpace G
inst✝¹¹ : LocallyCompactSpace G
inst✝¹⁰ : Group G
inst✝⁹ : TopologicalGroup G
inst✝⁸ : MeasurableSpace G
inst✝⁷ : BorelSpace G
μ' μ : Measure G
inst✝⁶ : IsFiniteMeasureOnCompacts μ
inst✝⁵ : IsFiniteMeasureOnCompacts μ'
inst✝⁴ : IsMulLeftInvariant μ
inst✝³ : IsMulLeftInvariant μ'
inst✝² : IsOpenPosMeasure μ
inst✝¹ : Regular μ
inst✝ : Regular μ'
c : ℝ≥0
hc : ∀ (s : Set G), ↑↑μ s < ⊤ → ↑↑μ' s < ⊤ → ↑↑μ' s = ↑↑(c • μ) s
⊢ ∀ (U : Set G), IsOpen U → ↑↑μ' U = ↑↑(c • μ) U

Next tactic:
intro U hU
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite
on compacts and regular, they coincide up to a multiplicative constant. -/
@[to_additive isAddLeftInvariant_eq_smul_of_regular]
lemma isMulLeftInvariant_eq_smul_of_regular
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [Regular μ] [Regular μ'] :
    ∃ (c : ℝ≥0), μ' = c • μ := by
  rcases measure_isMulLeftInvariant_eq_smul_of_ne_top μ' μ with ⟨c, hc⟩
  refine ⟨c, ?_⟩
  have A : ∀ U, IsOpen U → μ' U = (c • μ) U := by
    intro U hU
    

Proof state:
G : Type u_1
inst✝¹² : TopologicalSpace G
inst✝¹¹ : LocallyCompactSpace G
inst✝¹⁰ : Group G
inst✝⁹ : TopologicalGroup G
inst✝⁸ : MeasurableSpace G
inst✝⁷ : BorelSpace G
μ' μ : Measure G
inst✝⁶ : IsFiniteMeasureOnCompacts μ
inst✝⁵ : IsFiniteMeasureOnCompacts μ'
inst✝⁴ : IsMulLeftInvariant μ
inst✝³ : IsMulLeftInvariant μ'
inst✝² : IsOpenPosMeasure μ
inst✝¹ : Regular μ
inst✝ : Regular μ'
c : ℝ≥0
hc : ∀ (s : Set G), ↑↑μ s < ⊤ → ↑↑μ' s < ⊤ → ↑↑μ' s = ↑↑(c • μ) s
U : Set G
hU : IsOpen U
⊢ ↑↑μ' U = ↑↑(c • μ) U

Next tactic:
rw [hU.measure_eq_iSup_isCompact, hU.measure_eq_iSup_isCompact]
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite
on compacts and regular, they coincide up to a multiplicative constant. -/
@[to_additive isAddLeftInvariant_eq_smul_of_regular]
lemma isMulLeftInvariant_eq_smul_of_regular
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [Regular μ] [Regular μ'] :
    ∃ (c : ℝ≥0), μ' = c • μ := by
  rcases measure_isMulLeftInvariant_eq_smul_of_ne_top μ' μ with ⟨c, hc⟩
  refine ⟨c, ?_⟩
  have A : ∀ U, IsOpen U → μ' U = (c • μ) U := by
    intro U hU
    rw [hU.measure_eq_iSup_isCompact, hU.measure_eq_iSup_isCompact]
    

Proof state:
G : Type u_1
inst✝¹² : TopologicalSpace G
inst✝¹¹ : LocallyCompactSpace G
inst✝¹⁰ : Group G
inst✝⁹ : TopologicalGroup G
inst✝⁸ : MeasurableSpace G
inst✝⁷ : BorelSpace G
μ' μ : Measure G
inst✝⁶ : IsFiniteMeasureOnCompacts μ
inst✝⁵ : IsFiniteMeasureOnCompacts μ'
inst✝⁴ : IsMulLeftInvariant μ
inst✝³ : IsMulLeftInvariant μ'
inst✝² : IsOpenPosMeasure μ
inst✝¹ : Regular μ
inst✝ : Regular μ'
c : ℝ≥0
hc : ∀ (s : Set G), ↑↑μ s < ⊤ → ↑↑μ' s < ⊤ → ↑↑μ' s = ↑↑(c • μ) s
U : Set G
hU : IsOpen U
⊢ ⨆ K, ⨆ (_ : K ⊆ U), ⨆ (_ : IsCompact K), ↑↑μ' K = ⨆ K, ⨆ (_ : K ⊆ U), ⨆ (_ : IsCompact K), ↑↑(c • μ) K

Next tactic:
congr! 4 with K _KU K_comp
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite
on compacts and regular, they coincide up to a multiplicative constant. -/
@[to_additive isAddLeftInvariant_eq_smul_of_regular]
lemma isMulLeftInvariant_eq_smul_of_regular
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [Regular μ] [Regular μ'] :
    ∃ (c : ℝ≥0), μ' = c • μ := by
  rcases measure_isMulLeftInvariant_eq_smul_of_ne_top μ' μ with ⟨c, hc⟩
  refine ⟨c, ?_⟩
  have A : ∀ U, IsOpen U → μ' U = (c • μ) U := by
    intro U hU
    rw [hU.measure_eq_iSup_isCompact, hU.measure_eq_iSup_isCompact]
    congr! 4 with K _KU K_comp
    

Proof state:
case h.e'_4.h.f.f
G : Type u_1
inst✝¹² : TopologicalSpace G
inst✝¹¹ : LocallyCompactSpace G
inst✝¹⁰ : Group G
inst✝⁹ : TopologicalGroup G
inst✝⁸ : MeasurableSpace G
inst✝⁷ : BorelSpace G
μ' μ : Measure G
inst✝⁶ : IsFiniteMeasureOnCompacts μ
inst✝⁵ : IsFiniteMeasureOnCompacts μ'
inst✝⁴ : IsMulLeftInvariant μ
inst✝³ : IsMulLeftInvariant μ'
inst✝² : IsOpenPosMeasure μ
inst✝¹ : Regular μ
inst✝ : Regular μ'
c : ℝ≥0
hc : ∀ (s : Set G), ↑↑μ s < ⊤ → ↑↑μ' s < ⊤ → ↑↑μ' s = ↑↑(c • μ) s
U : Set G
hU : IsOpen U
K : Set G
_KU : K ⊆ U
K_comp : IsCompact K
⊢ ↑↑μ' K = ↑↑(c • μ) K

Next tactic:
exact hc K K_comp.measure_lt_top K_comp.measure_lt_top
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite
on compacts and regular, they coincide up to a multiplicative constant. -/
@[to_additive isAddLeftInvariant_eq_smul_of_regular]
lemma isMulLeftInvariant_eq_smul_of_regular
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [Regular μ] [Regular μ'] :
    ∃ (c : ℝ≥0), μ' = c • μ := by
  rcases measure_isMulLeftInvariant_eq_smul_of_ne_top μ' μ with ⟨c, hc⟩
  refine ⟨c, ?_⟩
  have A : ∀ U, IsOpen U → μ' U = (c • μ) U := by
    intro U hU
    rw [hU.measure_eq_iSup_isCompact, hU.measure_eq_iSup_isCompact]
    congr! 4 with K _KU K_comp
    exact hc K K_comp.measure_lt_top K_comp.measure_lt_top
  

Proof state:
case intro
G : Type u_1
inst✝¹² : TopologicalSpace G
inst✝¹¹ : LocallyCompactSpace G
inst✝¹⁰ : Group G
inst✝⁹ : TopologicalGroup G
inst✝⁸ : MeasurableSpace G
inst✝⁷ : BorelSpace G
μ' μ : Measure G
inst✝⁶ : IsFiniteMeasureOnCompacts μ
inst✝⁵ : IsFiniteMeasureOnCompacts μ'
inst✝⁴ : IsMulLeftInvariant μ
inst✝³ : IsMulLeftInvariant μ'
inst✝² : IsOpenPosMeasure μ
inst✝¹ : Regular μ
inst✝ : Regular μ'
c : ℝ≥0
hc : ∀ (s : Set G), ↑↑μ s < ⊤ → ↑↑μ' s < ⊤ → ↑↑μ' s = ↑↑(c • μ) s
A : ∀ (U : Set G), IsOpen U → ↑↑μ' U = ↑↑(c • μ) U
⊢ μ' = c • μ

Next tactic:
ext s _hs
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite
on compacts and regular, they coincide up to a multiplicative constant. -/
@[to_additive isAddLeftInvariant_eq_smul_of_regular]
lemma isMulLeftInvariant_eq_smul_of_regular
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [Regular μ] [Regular μ'] :
    ∃ (c : ℝ≥0), μ' = c • μ := by
  rcases measure_isMulLeftInvariant_eq_smul_of_ne_top μ' μ with ⟨c, hc⟩
  refine ⟨c, ?_⟩
  have A : ∀ U, IsOpen U → μ' U = (c • μ) U := by
    intro U hU
    rw [hU.measure_eq_iSup_isCompact, hU.measure_eq_iSup_isCompact]
    congr! 4 with K _KU K_comp
    exact hc K K_comp.measure_lt_top K_comp.measure_lt_top
  ext s _hs
  

Proof state:
case intro.h
G : Type u_1
inst✝¹² : TopologicalSpace G
inst✝¹¹ : LocallyCompactSpace G
inst✝¹⁰ : Group G
inst✝⁹ : TopologicalGroup G
inst✝⁸ : MeasurableSpace G
inst✝⁷ : BorelSpace G
μ' μ : Measure G
inst✝⁶ : IsFiniteMeasureOnCompacts μ
inst✝⁵ : IsFiniteMeasureOnCompacts μ'
inst✝⁴ : IsMulLeftInvariant μ
inst✝³ : IsMulLeftInvariant μ'
inst✝² : IsOpenPosMeasure μ
inst✝¹ : Regular μ
inst✝ : Regular μ'
c : ℝ≥0
hc : ∀ (s : Set G), ↑↑μ s < ⊤ → ↑↑μ' s < ⊤ → ↑↑μ' s = ↑↑(c • μ) s
A : ∀ (U : Set G), IsOpen U → ↑↑μ' U = ↑↑(c • μ) U
s : Set G
_hs : MeasurableSet s
⊢ ↑↑μ' s = ↑↑(c • μ) s

Next tactic:
rw [s.measure_eq_iInf_isOpen, s.measure_eq_iInf_isOpen]
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite
on compacts and regular, they coincide up to a multiplicative constant. -/
@[to_additive isAddLeftInvariant_eq_smul_of_regular]
lemma isMulLeftInvariant_eq_smul_of_regular
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [Regular μ] [Regular μ'] :
    ∃ (c : ℝ≥0), μ' = c • μ := by
  rcases measure_isMulLeftInvariant_eq_smul_of_ne_top μ' μ with ⟨c, hc⟩
  refine ⟨c, ?_⟩
  have A : ∀ U, IsOpen U → μ' U = (c • μ) U := by
    intro U hU
    rw [hU.measure_eq_iSup_isCompact, hU.measure_eq_iSup_isCompact]
    congr! 4 with K _KU K_comp
    exact hc K K_comp.measure_lt_top K_comp.measure_lt_top
  ext s _hs
  rw [s.measure_eq_iInf_isOpen, s.measure_eq_iInf_isOpen]
  

Proof state:
case intro.h
G : Type u_1
inst✝¹² : TopologicalSpace G
inst✝¹¹ : LocallyCompactSpace G
inst✝¹⁰ : Group G
inst✝⁹ : TopologicalGroup G
inst✝⁸ : MeasurableSpace G
inst✝⁷ : BorelSpace G
μ' μ : Measure G
inst✝⁶ : IsFiniteMeasureOnCompacts μ
inst✝⁵ : IsFiniteMeasureOnCompacts μ'
inst✝⁴ : IsMulLeftInvariant μ
inst✝³ : IsMulLeftInvariant μ'
inst✝² : IsOpenPosMeasure μ
inst✝¹ : Regular μ
inst✝ : Regular μ'
c : ℝ≥0
hc : ∀ (s : Set G), ↑↑μ s < ⊤ → ↑↑μ' s < ⊤ → ↑↑μ' s = ↑↑(c • μ) s
A : ∀ (U : Set G), IsOpen U → ↑↑μ' U = ↑↑(c • μ) U
s : Set G
_hs : MeasurableSet s
⊢ ⨅ U, ⨅ (_ : s ⊆ U), ⨅ (_ : IsOpen U), ↑↑μ' U = ⨅ U, ⨅ (_ : s ⊆ U), ⨅ (_ : IsOpen U), ↑↑(c • μ) U

Next tactic:
congr! 4 with U _sU U_open