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

Proof steps:
state_before: G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
inst✝⁴ : LocallyCompactSpace G
μ' μ : Measure G
inst✝³ : μ.IsHaarMeasure
inst✝² : IsFiniteMeasureOnCompacts μ'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : μ.InnerRegularCompactLTTop
s : Set G
hs : MeasurableSet s
h's : IsCompact (closure s)
⊢ μ'.haarScalarFactor μ • μ s ≤ μ' s
state_after: G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
inst✝⁴ : LocallyCompactSpace G
μ' μ : Measure G
inst✝³ : μ.IsHaarMeasure
inst✝² : IsFiniteMeasureOnCompacts μ'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : μ.InnerRegularCompactLTTop
s : Set G
hs : MeasurableSet s
h's : IsCompact (closure s)
r : ℝ≥0∞
hr : r < μ'.haarScalarFactor μ • μ s
⊢ r < μ' s
tactic: apply le_of_forall_lt (fun r hr ↦ ?_)
state_before: G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
inst✝⁴ : LocallyCompactSpace G
μ' μ : Measure G
inst✝³ : μ.IsHaarMeasure
inst✝² : IsFiniteMeasureOnCompacts μ'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : μ.InnerRegularCompactLTTop
s : Set G
hs : MeasurableSet s
h's : IsCompact (closure s)
r : ℝ≥0∞
hr : r < μ'.haarScalarFactor μ • μ s
⊢ r < μ' s
state_after: G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
inst✝⁴ : LocallyCompactSpace G
μ' μ : Measure G
inst✝³ : μ.IsHaarMeasure
inst✝² : IsFiniteMeasureOnCompacts μ'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : μ.InnerRegularCompactLTTop
s : Set G
hs : MeasurableSet s
h's : IsCompact (closure s)
r : ℝ≥0∞
hr : r < μ'.haarScalarFactor μ • μ s
ν : Measure G := μ'.haarScalarFactor μ • μ
⊢ r < μ' s
tactic: let ν := haarScalarFactor μ' μ • μ
state_before: G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
inst✝⁴ : LocallyCompactSpace G
μ' μ : Measure G
inst✝³ : μ.IsHaarMeasure
inst✝² : IsFiniteMeasureOnCompacts μ'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : μ.InnerRegularCompactLTTop
s : Set G
hs : MeasurableSet s
h's : IsCompact (closure s)
r : ℝ≥0∞
hr : r < μ'.haarScalarFactor μ • μ s
ν : Measure G := μ'.haarScalarFactor μ • μ
⊢ r < μ' s
state_after: G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
inst✝⁴ : LocallyCompactSpace G
μ' μ : Measure G
inst✝³ : μ.IsHaarMeasure
inst✝² : IsFiniteMeasureOnCompacts μ'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : μ.InnerRegularCompactLTTop
s : Set G
hs : MeasurableSet s
h's : IsCompact (closure s)
r : ℝ≥0∞
hr : r < μ'.haarScalarFactor μ • μ s
ν : Measure G := μ'.haarScalarFactor μ • μ
this : ν s ≠ ⊤
⊢ r < μ' s
tactic: have : ν s ≠ ∞ := ((measure_mono subset_closure).trans_lt h's.measure_lt_top).ne
state_before: G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
inst✝⁴ : LocallyCompactSpace G
μ' μ : Measure G
inst✝³ : μ.IsHaarMeasure
inst✝² : IsFiniteMeasureOnCompacts μ'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : μ.InnerRegularCompactLTTop
s : Set G
hs : MeasurableSet s
h's : IsCompact (closure s)
r : ℝ≥0∞
hr : r < μ'.haarScalarFactor μ • μ s
ν : Measure G := μ'.haarScalarFactor μ • μ
this : ν s ≠ ⊤
⊢ r < μ' s
state_after: case intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
inst✝⁴ : LocallyCompactSpace G
μ' μ : Measure G
inst✝³ : μ.IsHaarMeasure
inst✝² : IsFiniteMeasureOnCompacts μ'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : μ.InnerRegularCompactLTTop
s : Set G
hs : MeasurableSet s
h's : IsCompact (closure s)
r : ℝ≥0∞
hr : r < μ'.haarScalarFactor μ • μ s
ν : Measure G := μ'.haarScalarFactor μ • μ
this : ν s ≠ ⊤
f : G → ℝ
f_cont : Continuous f
f_comp : HasCompactSupport f
hf : f ⁻¹' {1} ⊆ s
νf : r < ν (f ⁻¹' {1})
⊢ r < μ' s
tactic: obtain ⟨-, hf, ⟨f, f_cont, f_comp, rfl⟩, νf⟩ :
      ∃ K ⊆ s, (∃ f, Continuous f ∧ HasCompactSupport f ∧ K = f ⁻¹' {1}) ∧ r < ν K :=
    innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_group ⟨hs, this⟩ r
      (by convert hr)
state_before: case intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
inst✝⁴ : LocallyCompactSpace G
μ' μ : Measure G
inst✝³ : μ.IsHaarMeasure
inst✝² : IsFiniteMeasureOnCompacts μ'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : μ.InnerRegularCompactLTTop
s : Set G
hs : MeasurableSet s
h's : IsCompact (closure s)
r : ℝ≥0∞
hr : r < μ'.haarScalarFactor μ • μ s
ν : Measure G := μ'.haarScalarFactor μ • μ
this : ν s ≠ ⊤
f : G → ℝ
f_cont : Continuous f
f_comp : HasCompactSupport f
hf : f ⁻¹' {1} ⊆ s
νf : r < ν (f ⁻¹' {1})
⊢ r < μ' s
state_after: no goals
tactic: calc
  r < ν (f ⁻¹' {1}) := νf
  _ = μ' (f ⁻¹' {1}) :=
    (measure_preimage_isMulLeftInvariant_eq_smul_of_hasCompactSupport _ _ f_cont f_comp).symm
  _ ≤ μ' s := measure_mono hf