l3lab/ntp-mathlib huggingface dataset:
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite on
compacts and inner regular for finite measure sets with respect to compact sets,
they coincide in the following sense: they give the same value to finite measure sets,
up to a multiplicative constant. -/
@[to_additive]
lemma measure_isMulLeftInvariant_eq_smul_of_ne_top
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [InnerRegularCompactLTTop μ] [InnerRegularCompactLTTop μ'] :
    ∃ (c : ℝ≥0), ∀ (s : Set G), μ s < ∞ → μ' s < ∞ → μ' s = (c • μ) s := by
  /- We know that the measures integrate in the same way continuous compactly supported functions,
  up to a constant `c`. We will use this constant `c`. -/
  rcases integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport μ' μ with ⟨c, hc⟩
  refine ⟨c, fun s hs h's ↦ ?_⟩
  /- By regularity, every compact set may be approximated by a continuous compactly supported
  function. Therefore, the measures coincide on compact sets. -/
  have A : ∀ k, IsCompact k → μ' k = (c • μ) k := by
    intro k hk
    rw [hk.measure_eq_biInf_integral_hasCompactSupport μ',
        hk.measure_eq_biInf_integral_hasCompactSupport (c • μ)]
    

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 : InnerRegularCompactLTTop μ
inst✝ : InnerRegularCompactLTTop μ'
c : ℝ≥0
hc : ∀ (f : G → ℝ), Continuous f → HasCompactSupport f → ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂c • μ
s : Set G
hs : ↑↑μ s < ⊤
h's : ↑↑μ' s < ⊤
k : Set G
hk : IsCompact k
⊢ ⨅ f,
      ⨅ (_ : Continuous f),
        ⨅ (_ : HasCompactSupport f), ⨅ (_ : EqOn f 1 k), ⨅ (_ : 0 ≤ f), ENNReal.ofReal (∫ (x : G), f x ∂μ') =
    ⨅ f,
      ⨅ (_ : Continuous f),
        ⨅ (_ : HasCompactSupport f), ⨅ (_ : EqOn f 1 k), ⨅ (_ : 0 ≤ f), ENNReal.ofReal (∫ (x : G), f x ∂c • μ)

Next tactic:
congr! 7 with f f_cont f_comp _fk _f_nonneg
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite on
compacts and inner regular for finite measure sets with respect to compact sets,
they coincide in the following sense: they give the same value to finite measure sets,
up to a multiplicative constant. -/
@[to_additive]
lemma measure_isMulLeftInvariant_eq_smul_of_ne_top
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [InnerRegularCompactLTTop μ] [InnerRegularCompactLTTop μ'] :
    ∃ (c : ℝ≥0), ∀ (s : Set G), μ s < ∞ → μ' s < ∞ → μ' s = (c • μ) s := by
  /- We know that the measures integrate in the same way continuous compactly supported functions,
  up to a constant `c`. We will use this constant `c`. -/
  rcases integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport μ' μ with ⟨c, hc⟩
  refine ⟨c, fun s hs h's ↦ ?_⟩
  /- By regularity, every compact set may be approximated by a continuous compactly supported
  function. Therefore, the measures coincide on compact sets. -/
  have A : ∀ k, IsCompact k → μ' k = (c • μ) k := by
    intro k hk
    rw [hk.measure_eq_biInf_integral_hasCompactSupport μ',
        hk.measure_eq_biInf_integral_hasCompactSupport (c • μ)]
    congr! 7 with f f_cont f_comp _fk _f_nonneg
    

Proof state:
case h.e'_4.h.f.f.f.f.h.e'_1
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 : InnerRegularCompactLTTop μ
inst✝ : InnerRegularCompactLTTop μ'
c : ℝ≥0
hc : ∀ (f : G → ℝ), Continuous f → HasCompactSupport f → ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂c • μ
s : Set G
hs : ↑↑μ s < ⊤
h's : ↑↑μ' s < ⊤
k : Set G
hk : IsCompact k
f : G → ℝ
f_cont : Continuous f
f_comp : HasCompactSupport f
_fk : EqOn f 1 k
_f_nonneg : 0 ≤ f
⊢ ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂c • μ

Next tactic:
exact hc f f_cont f_comp
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite on
compacts and inner regular for finite measure sets with respect to compact sets,
they coincide in the following sense: they give the same value to finite measure sets,
up to a multiplicative constant. -/
@[to_additive]
lemma measure_isMulLeftInvariant_eq_smul_of_ne_top
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [InnerRegularCompactLTTop μ] [InnerRegularCompactLTTop μ'] :
    ∃ (c : ℝ≥0), ∀ (s : Set G), μ s < ∞ → μ' s < ∞ → μ' s = (c • μ) s := by
  /- We know that the measures integrate in the same way continuous compactly supported functions,
  up to a constant `c`. We will use this constant `c`. -/
  rcases integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport μ' μ with ⟨c, hc⟩
  refine ⟨c, fun s hs h's ↦ ?_⟩
  /- By regularity, every compact set may be approximated by a continuous compactly supported
  function. Therefore, the measures coincide on compact sets. -/
  have A : ∀ k, IsCompact k → μ' k = (c • μ) k := by
    intro k hk
    rw [hk.measure_eq_biInf_integral_hasCompactSupport μ',
        hk.measure_eq_biInf_integral_hasCompactSupport (c • μ)]
    congr! 7 with f f_cont f_comp _fk _f_nonneg
    exact hc f f_cont f_comp
  /- By regularity, every measurable set of finite measure may be approximated by compact sets.
  Therefore, the measures coincide on measurable sets of finite measure. -/
  

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 : InnerRegularCompactLTTop μ
inst✝ : InnerRegularCompactLTTop μ'
c : ℝ≥0
hc : ∀ (f : G → ℝ), Continuous f → HasCompactSupport f → ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂c • μ
s : Set G
hs : ↑↑μ s < ⊤
h's : ↑↑μ' s < ⊤
A : ∀ (k : Set G), IsCompact k → ↑↑μ' k = ↑↑(c • μ) k
⊢ ↑↑μ' s = ↑↑(c • μ) s

Next tactic:
have B : ∀ s, MeasurableSet s → μ s < ∞ → μ' s < ∞ → μ' s = (c • μ) s := by
    intro s s_meas hs h's
    have : (c • μ) s ≠ ∞ := by simp [ENNReal.mul_eq_top, hs.ne]
    rw [s_meas.measure_eq_iSup_isCompact_of_ne_top h's.ne,
        s_meas.measure_eq_iSup_isCompact_of_ne_top this]
    congr! 4 with K _Ks K_comp
    exact A K K_comp
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite on
compacts and inner regular for finite measure sets with respect to compact sets,
they coincide in the following sense: they give the same value to finite measure sets,
up to a multiplicative constant. -/
@[to_additive]
lemma measure_isMulLeftInvariant_eq_smul_of_ne_top
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [InnerRegularCompactLTTop μ] [InnerRegularCompactLTTop μ'] :
    ∃ (c : ℝ≥0), ∀ (s : Set G), μ s < ∞ → μ' s < ∞ → μ' s = (c • μ) s := by
  /- We know that the measures integrate in the same way continuous compactly supported functions,
  up to a constant `c`. We will use this constant `c`. -/
  rcases integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport μ' μ with ⟨c, hc⟩
  refine ⟨c, fun s hs h's ↦ ?_⟩
  /- By regularity, every compact set may be approximated by a continuous compactly supported
  function. Therefore, the measures coincide on compact sets. -/
  have A : ∀ k, IsCompact k → μ' k = (c • μ) k := by
    intro k hk
    rw [hk.measure_eq_biInf_integral_hasCompactSupport μ',
        hk.measure_eq_biInf_integral_hasCompactSupport (c • μ)]
    congr! 7 with f f_cont f_comp _fk _f_nonneg
    exact hc f f_cont f_comp
  /- By regularity, every measurable set of finite measure may be approximated by compact sets.
  Therefore, the measures coincide on measurable sets of finite measure. -/
  have B : ∀ s, MeasurableSet s → μ s < ∞ → μ' s < ∞ → μ' s = (c • μ) s := 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 : InnerRegularCompactLTTop μ
inst✝ : InnerRegularCompactLTTop μ'
c : ℝ≥0
hc : ∀ (f : G → ℝ), Continuous f → HasCompactSupport f → ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂c • μ
s : Set G
hs : ↑↑μ s < ⊤
h's : ↑↑μ' s < ⊤
A : ∀ (k : Set G), IsCompact k → ↑↑μ' k = ↑↑(c • μ) k
⊢ ∀ (s : Set G), MeasurableSet s → ↑↑μ s < ⊤ → ↑↑μ' s < ⊤ → ↑↑μ' s = ↑↑(c • μ) s

Next tactic:
intro s s_meas hs h's
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite on
compacts and inner regular for finite measure sets with respect to compact sets,
they coincide in the following sense: they give the same value to finite measure sets,
up to a multiplicative constant. -/
@[to_additive]
lemma measure_isMulLeftInvariant_eq_smul_of_ne_top
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [InnerRegularCompactLTTop μ] [InnerRegularCompactLTTop μ'] :
    ∃ (c : ℝ≥0), ∀ (s : Set G), μ s < ∞ → μ' s < ∞ → μ' s = (c • μ) s := by
  /- We know that the measures integrate in the same way continuous compactly supported functions,
  up to a constant `c`. We will use this constant `c`. -/
  rcases integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport μ' μ with ⟨c, hc⟩
  refine ⟨c, fun s hs h's ↦ ?_⟩
  /- By regularity, every compact set may be approximated by a continuous compactly supported
  function. Therefore, the measures coincide on compact sets. -/
  have A : ∀ k, IsCompact k → μ' k = (c • μ) k := by
    intro k hk
    rw [hk.measure_eq_biInf_integral_hasCompactSupport μ',
        hk.measure_eq_biInf_integral_hasCompactSupport (c • μ)]
    congr! 7 with f f_cont f_comp _fk _f_nonneg
    exact hc f f_cont f_comp
  /- By regularity, every measurable set of finite measure may be approximated by compact sets.
  Therefore, the measures coincide on measurable sets of finite measure. -/
  have B : ∀ s, MeasurableSet s → μ s < ∞ → μ' s < ∞ → μ' s = (c • μ) s := by
    intro s s_meas hs h's
    

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 : InnerRegularCompactLTTop μ
inst✝ : InnerRegularCompactLTTop μ'
c : ℝ≥0
hc : ∀ (f : G → ℝ), Continuous f → HasCompactSupport f → ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂c • μ
s✝ : Set G
hs✝ : ↑↑μ s✝ < ⊤
h's✝ : ↑↑μ' s✝ < ⊤
A : ∀ (k : Set G), IsCompact k → ↑↑μ' k = ↑↑(c • μ) k
s : Set G
s_meas : MeasurableSet s
hs : ↑↑μ s < ⊤
h's : ↑↑μ' s < ⊤
⊢ ↑↑μ' s = ↑↑(c • μ) s

Next tactic:
have : (c • μ) s ≠ ∞ := by simp [ENNReal.mul_eq_top, hs.ne]
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite on
compacts and inner regular for finite measure sets with respect to compact sets,
they coincide in the following sense: they give the same value to finite measure sets,
up to a multiplicative constant. -/
@[to_additive]
lemma measure_isMulLeftInvariant_eq_smul_of_ne_top
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [InnerRegularCompactLTTop μ] [InnerRegularCompactLTTop μ'] :
    ∃ (c : ℝ≥0), ∀ (s : Set G), μ s < ∞ → μ' s < ∞ → μ' s = (c • μ) s := by
  /- We know that the measures integrate in the same way continuous compactly supported functions,
  up to a constant `c`. We will use this constant `c`. -/
  rcases integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport μ' μ with ⟨c, hc⟩
  refine ⟨c, fun s hs h's ↦ ?_⟩
  /- By regularity, every compact set may be approximated by a continuous compactly supported
  function. Therefore, the measures coincide on compact sets. -/
  have A : ∀ k, IsCompact k → μ' k = (c • μ) k := by
    intro k hk
    rw [hk.measure_eq_biInf_integral_hasCompactSupport μ',
        hk.measure_eq_biInf_integral_hasCompactSupport (c • μ)]
    congr! 7 with f f_cont f_comp _fk _f_nonneg
    exact hc f f_cont f_comp
  /- By regularity, every measurable set of finite measure may be approximated by compact sets.
  Therefore, the measures coincide on measurable sets of finite measure. -/
  have B : ∀ s, MeasurableSet s → μ s < ∞ → μ' s < ∞ → μ' s = (c • μ) s := by
    intro s s_meas hs h's
    have : (c • μ) s ≠ ∞ := 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 : InnerRegularCompactLTTop μ
inst✝ : InnerRegularCompactLTTop μ'
c : ℝ≥0
hc : ∀ (f : G → ℝ), Continuous f → HasCompactSupport f → ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂c • μ
s✝ : Set G
hs✝ : ↑↑μ s✝ < ⊤
h's✝ : ↑↑μ' s✝ < ⊤
A : ∀ (k : Set G), IsCompact k → ↑↑μ' k = ↑↑(c • μ) k
s : Set G
s_meas : MeasurableSet s
hs : ↑↑μ s < ⊤
h's : ↑↑μ' s < ⊤
⊢ ↑↑(c • μ) s ≠ ⊤

Next tactic:
simp [ENNReal.mul_eq_top, hs.ne]
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite on
compacts and inner regular for finite measure sets with respect to compact sets,
they coincide in the following sense: they give the same value to finite measure sets,
up to a multiplicative constant. -/
@[to_additive]
lemma measure_isMulLeftInvariant_eq_smul_of_ne_top
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [InnerRegularCompactLTTop μ] [InnerRegularCompactLTTop μ'] :
    ∃ (c : ℝ≥0), ∀ (s : Set G), μ s < ∞ → μ' s < ∞ → μ' s = (c • μ) s := by
  /- We know that the measures integrate in the same way continuous compactly supported functions,
  up to a constant `c`. We will use this constant `c`. -/
  rcases integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport μ' μ with ⟨c, hc⟩
  refine ⟨c, fun s hs h's ↦ ?_⟩
  /- By regularity, every compact set may be approximated by a continuous compactly supported
  function. Therefore, the measures coincide on compact sets. -/
  have A : ∀ k, IsCompact k → μ' k = (c • μ) k := by
    intro k hk
    rw [hk.measure_eq_biInf_integral_hasCompactSupport μ',
        hk.measure_eq_biInf_integral_hasCompactSupport (c • μ)]
    congr! 7 with f f_cont f_comp _fk _f_nonneg
    exact hc f f_cont f_comp
  /- By regularity, every measurable set of finite measure may be approximated by compact sets.
  Therefore, the measures coincide on measurable sets of finite measure. -/
  have B : ∀ s, MeasurableSet s → μ s < ∞ → μ' s < ∞ → μ' s = (c • μ) s := by
    intro s s_meas hs h's
    have : (c • μ) s ≠ ∞ := by simp [ENNReal.mul_eq_top, hs.ne]
    

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 : InnerRegularCompactLTTop μ
inst✝ : InnerRegularCompactLTTop μ'
c : ℝ≥0
hc : ∀ (f : G → ℝ), Continuous f → HasCompactSupport f → ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂c • μ
s✝ : Set G
hs✝ : ↑↑μ s✝ < ⊤
h's✝ : ↑↑μ' s✝ < ⊤
A : ∀ (k : Set G), IsCompact k → ↑↑μ' k = ↑↑(c • μ) k
s : Set G
s_meas : MeasurableSet s
hs : ↑↑μ s < ⊤
h's : ↑↑μ' s < ⊤
this : ↑↑(c • μ) s ≠ ⊤
⊢ ↑↑μ' s = ↑↑(c • μ) s

Next tactic:
rw [s_meas.measure_eq_iSup_isCompact_of_ne_top h's.ne,
        s_meas.measure_eq_iSup_isCompact_of_ne_top this]
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite on
compacts and inner regular for finite measure sets with respect to compact sets,
they coincide in the following sense: they give the same value to finite measure sets,
up to a multiplicative constant. -/
@[to_additive]
lemma measure_isMulLeftInvariant_eq_smul_of_ne_top
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [InnerRegularCompactLTTop μ] [InnerRegularCompactLTTop μ'] :
    ∃ (c : ℝ≥0), ∀ (s : Set G), μ s < ∞ → μ' s < ∞ → μ' s = (c • μ) s := by
  /- We know that the measures integrate in the same way continuous compactly supported functions,
  up to a constant `c`. We will use this constant `c`. -/
  rcases integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport μ' μ with ⟨c, hc⟩
  refine ⟨c, fun s hs h's ↦ ?_⟩
  /- By regularity, every compact set may be approximated by a continuous compactly supported
  function. Therefore, the measures coincide on compact sets. -/
  have A : ∀ k, IsCompact k → μ' k = (c • μ) k := by
    intro k hk
    rw [hk.measure_eq_biInf_integral_hasCompactSupport μ',
        hk.measure_eq_biInf_integral_hasCompactSupport (c • μ)]
    congr! 7 with f f_cont f_comp _fk _f_nonneg
    exact hc f f_cont f_comp
  /- By regularity, every measurable set of finite measure may be approximated by compact sets.
  Therefore, the measures coincide on measurable sets of finite measure. -/
  have B : ∀ s, MeasurableSet s → μ s < ∞ → μ' s < ∞ → μ' s = (c • μ) s := by
    intro s s_meas hs h's
    have : (c • μ) s ≠ ∞ := by simp [ENNReal.mul_eq_top, hs.ne]
    rw [s_meas.measure_eq_iSup_isCompact_of_ne_top h's.ne,
        s_meas.measure_eq_iSup_isCompact_of_ne_top this]
    

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 : InnerRegularCompactLTTop μ
inst✝ : InnerRegularCompactLTTop μ'
c : ℝ≥0
hc : ∀ (f : G → ℝ), Continuous f → HasCompactSupport f → ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂c • μ
s✝ : Set G
hs✝ : ↑↑μ s✝ < ⊤
h's✝ : ↑↑μ' s✝ < ⊤
A : ∀ (k : Set G), IsCompact k → ↑↑μ' k = ↑↑(c • μ) k
s : Set G
s_meas : MeasurableSet s
hs : ↑↑μ s < ⊤
h's : ↑↑μ' s < ⊤
this : ↑↑(c • μ) 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 for finite measure sets with respect to compact sets,
they coincide in the following sense: they give the same value to finite measure sets,
up to a multiplicative constant. -/
@[to_additive]
lemma measure_isMulLeftInvariant_eq_smul_of_ne_top
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ]
    [InnerRegularCompactLTTop μ] [InnerRegularCompactLTTop μ'] :
    ∃ (c : ℝ≥0), ∀ (s : Set G), μ s < ∞ → μ' s < ∞ → μ' s = (c • μ) s := by
  /- We know that the measures integrate in the same way continuous compactly supported functions,
  up to a constant `c`. We will use this constant `c`. -/
  rcases integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport μ' μ with ⟨c, hc⟩
  refine ⟨c, fun s hs h's ↦ ?_⟩
  /- By regularity, every compact set may be approximated by a continuous compactly supported
  function. Therefore, the measures coincide on compact sets. -/
  have A : ∀ k, IsCompact k → μ' k = (c • μ) k := by
    intro k hk
    rw [hk.measure_eq_biInf_integral_hasCompactSupport μ',
        hk.measure_eq_biInf_integral_hasCompactSupport (c • μ)]
    congr! 7 with f f_cont f_comp _fk _f_nonneg
    exact hc f f_cont f_comp
  /- By regularity, every measurable set of finite measure may be approximated by compact sets.
  Therefore, the measures coincide on measurable sets of finite measure. -/
  have B : ∀ s, MeasurableSet s → μ s < ∞ → μ' s < ∞ → μ' s = (c • μ) s := by
    intro s s_meas hs h's
    have : (c • μ) s ≠ ∞ := by simp [ENNReal.mul_eq_top, hs.ne]
    rw [s_meas.measure_eq_iSup_isCompact_of_ne_top h's.ne,
        s_meas.measure_eq_iSup_isCompact_of_ne_top this]
    congr! 4 with K _Ks 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 : InnerRegularCompactLTTop μ
inst✝ : InnerRegularCompactLTTop μ'
c : ℝ≥0
hc : ∀ (f : G → ℝ), Continuous f → HasCompactSupport f → ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂c • μ
s✝ : Set G
hs✝ : ↑↑μ s✝ < ⊤
h's✝ : ↑↑μ' s✝ < ⊤
A : ∀ (k : Set G), IsCompact k → ↑↑μ' k = ↑↑(c • μ) k
s : Set G
s_meas : MeasurableSet s
hs : ↑↑μ s < ⊤
h's : ↑↑μ' s < ⊤
this : ↑↑(c • μ) s ≠ ⊤
K : Set G
_Ks : K ⊆ s
K_comp : IsCompact K
⊢ ↑↑μ' K = ↑↑(c • μ) K

Next tactic:
exact A K K_comp