Theorem: MeasureTheory.Measure.isMulLeftInvariant_eq_smul_of_regular
File: Mathlib/MeasureTheory/Measure/Haar/Unique.lean

Proof steps:
state_before: G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : Group G
inst✝⁸ : TopologicalGroup G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : LocallyCompactSpace G
μ' μ : Measure G
inst✝⁴ : μ.IsHaarMeasure
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : μ'.IsMulLeftInvariant
inst✝¹ : μ.Regular
inst✝ : μ'.Regular
⊢ μ' = μ'.haarScalarFactor μ • μ
state_after: G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : Group G
inst✝⁸ : TopologicalGroup G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : LocallyCompactSpace G
μ' μ : Measure G
inst✝⁴ : μ.IsHaarMeasure
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : μ'.IsMulLeftInvariant
inst✝¹ : μ.Regular
inst✝ : μ'.Regular
A : ∀ (U : Set G), IsOpen U → μ' U = (μ'.haarScalarFactor μ • μ) U
⊢ μ' = μ'.haarScalarFactor μ • μ
tactic: have A : ∀ U, IsOpen U → μ' U = (haarScalarFactor μ' μ • 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 measure_isMulLeftInvariant_eq_smul_of_ne_top μ' μ K_comp.measure_lt_top.ne
      K_comp.measure_lt_top.ne
state_before: G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : Group G
inst✝⁸ : TopologicalGroup G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : LocallyCompactSpace G
μ' μ : Measure G
inst✝⁴ : μ.IsHaarMeasure
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : μ'.IsMulLeftInvariant
inst✝¹ : μ.Regular
inst✝ : μ'.Regular
A : ∀ (U : Set G), IsOpen U → μ' U = (μ'.haarScalarFactor μ • μ) U
⊢ μ' = μ'.haarScalarFactor μ • μ
state_after: case h
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : Group G
inst✝⁸ : TopologicalGroup G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : LocallyCompactSpace G
μ' μ : Measure G
inst✝⁴ : μ.IsHaarMeasure
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : μ'.IsMulLeftInvariant
inst✝¹ : μ.Regular
inst✝ : μ'.Regular
A : ∀ (U : Set G), IsOpen U → μ' U = (μ'.haarScalarFactor μ • μ) U
s : Set G
_hs : MeasurableSet s
⊢ μ' s = (μ'.haarScalarFactor μ • μ) s
tactic: ext s _hs
state_before: case h
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : Group G
inst✝⁸ : TopologicalGroup G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : LocallyCompactSpace G
μ' μ : Measure G
inst✝⁴ : μ.IsHaarMeasure
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : μ'.IsMulLeftInvariant
inst✝¹ : μ.Regular
inst✝ : μ'.Regular
A : ∀ (U : Set G), IsOpen U → μ' U = (μ'.haarScalarFactor μ • μ) U
s : Set G
_hs : MeasurableSet s
⊢ μ' s = (μ'.haarScalarFactor μ • μ) s
state_after: case h
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : Group G
inst✝⁸ : TopologicalGroup G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : LocallyCompactSpace G
μ' μ : Measure G
inst✝⁴ : μ.IsHaarMeasure
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : μ'.IsMulLeftInvariant
inst✝¹ : μ.Regular
inst✝ : μ'.Regular
A : ∀ (U : Set G), IsOpen U → μ' U = (μ'.haarScalarFactor μ • μ) U
s : Set G
_hs : MeasurableSet s
⊢ ⨅ U, ⨅ (_ : s ⊆ U), ⨅ (_ : IsOpen U), μ' U = ⨅ U, ⨅ (_ : s ⊆ U), ⨅ (_ : IsOpen U), (μ'.haarScalarFactor μ • μ) U
tactic: rw [s.measure_eq_iInf_isOpen, s.measure_eq_iInf_isOpen]
state_before: case h
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : Group G
inst✝⁸ : TopologicalGroup G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : LocallyCompactSpace G
μ' μ : Measure G
inst✝⁴ : μ.IsHaarMeasure
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : μ'.IsMulLeftInvariant
inst✝¹ : μ.Regular
inst✝ : μ'.Regular
A : ∀ (U : Set G), IsOpen U → μ' U = (μ'.haarScalarFactor μ • μ) U
s : Set G
_hs : MeasurableSet s
⊢ ⨅ U, ⨅ (_ : s ⊆ U), ⨅ (_ : IsOpen U), μ' U = ⨅ U, ⨅ (_ : s ⊆ U), ⨅ (_ : IsOpen U), (μ'.haarScalarFactor μ • μ) U
state_after: case h.h.e'_4.h.f.f
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : Group G
inst✝⁸ : TopologicalGroup G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : LocallyCompactSpace G
μ' μ : Measure G
inst✝⁴ : μ.IsHaarMeasure
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : μ'.IsMulLeftInvariant
inst✝¹ : μ.Regular
inst✝ : μ'.Regular
A : ∀ (U : Set G), IsOpen U → μ' U = (μ'.haarScalarFactor μ • μ) U
s : Set G
_hs : MeasurableSet s
U : Set G
_sU : s ⊆ U
U_open : IsOpen U
⊢ μ' U = (μ'.haarScalarFactor μ • μ) U
tactic: congr! 4 with U _sU U_open
state_before: case h.h.e'_4.h.f.f
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : Group G
inst✝⁸ : TopologicalGroup G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : LocallyCompactSpace G
μ' μ : Measure G
inst✝⁴ : μ.IsHaarMeasure
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : μ'.IsMulLeftInvariant
inst✝¹ : μ.Regular
inst✝ : μ'.Regular
A : ∀ (U : Set G), IsOpen U → μ' U = (μ'.haarScalarFactor μ • μ) U
s : Set G
_hs : MeasurableSet s
U : Set G
_sU : s ⊆ U
U_open : IsOpen U
⊢ μ' U = (μ'.haarScalarFactor μ • μ) U
state_after: no goals
tactic: exact A U U_open