/-- **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
    exact A K K_comp
  /- Finally, replace an arbitrary finite measure set with a measurable version, and use the
  version for measurable sets. -/
  let t := toMeasurable μ' s ∩ toMeasurable μ s
  have st : s ⊆ t := subset_inter (subset_toMeasurable μ' s) (subset_toMeasurable μ s)
  have mu'_t : μ' t = μ' s := by
    apply le_antisymm
    · exact (measure_mono (inter_subset_left _ _)).trans (measure_toMeasurable s).le
    · exact measure_mono st
  have mu_t : μ t = μ s := by
    apply le_antisymm
    · exact (measure_mono (inter_subset_right _ _)).trans (measure_toMeasurable s).le
    · exact measure_mono st
  

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
B : ∀ (s : Set G), MeasurableSet s → ↑↑μ s < ⊤ → ↑↑μ' s < ⊤ → ↑↑μ' s = ↑↑(c • μ) s
t : Set G := toMeasurable μ' s ∩ toMeasurable μ s
st : s ⊆ t
mu'_t : ↑↑μ' t = ↑↑μ' s
mu_t : ↑↑μ t = ↑↑μ s
⊢ ↑↑μ' s = ↑↑(c • μ) s

Next tactic:
simp only [← mu'_t, smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply, ← mu_t,
    nnreal_smul_coe_apply]