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

Proof steps:
state_before: G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
s : Set G
hs : MeasurableSet s
h's : IsCompact (closure s)
⊢ μ' s = μ'.haarScalarFactor μ • μ s
state_after: G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
s : Set G
hs : MeasurableSet s
h's : IsCompact (closure s)
ν : Measure G := haar
⊢ μ' s = μ'.haarScalarFactor μ • μ s
tactic: let ν : Measure G := haar
state_before: G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
s : Set G
hs : MeasurableSet s
h's : IsCompact (closure s)
ν : Measure G := haar
⊢ μ' s = μ'.haarScalarFactor μ • μ s
state_after: G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
s : Set G
hs : MeasurableSet s
h's : IsCompact (closure s)
ν : Measure G := haar
A : μ' s = μ'.haarScalarFactor ν • ν s
⊢ μ' s = μ'.haarScalarFactor μ • μ s
tactic: have A : μ' s = haarScalarFactor μ' ν • ν s :=
    measure_isMulInvariant_eq_smul_of_isCompact_closure_of_innerRegularCompactLTTop μ' ν hs h's
state_before: G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
s : Set G
hs : MeasurableSet s
h's : IsCompact (closure s)
ν : Measure G := haar
A : μ' s = μ'.haarScalarFactor ν • ν s
⊢ μ' s = μ'.haarScalarFactor μ • μ s
state_after: G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
s : Set G
hs : MeasurableSet s
h's : IsCompact (closure s)
ν : Measure G := haar
A : μ' s = μ'.haarScalarFactor ν • ν s
B : μ s = μ.haarScalarFactor ν • ν s
⊢ μ' s = μ'.haarScalarFactor μ • μ s
tactic: have B : μ s = haarScalarFactor μ ν • ν s :=
    measure_isMulInvariant_eq_smul_of_isCompact_closure_of_innerRegularCompactLTTop μ ν hs h's
state_before: G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
s : Set G
hs : MeasurableSet s
h's : IsCompact (closure s)
ν : Measure G := haar
A : μ' s = μ'.haarScalarFactor ν • ν s
B : μ s = μ.haarScalarFactor ν • ν s
⊢ μ' s = μ'.haarScalarFactor μ • μ s
state_after: no goals
tactic: rw [A, B, smul_smul, haarScalarFactor_eq_mul μ' μ ν]