-- Mathlib/MeasureTheory/Group/Measure.lean
theorem measure_univ_of_isMulLeftInvariant [WeaklyLocallyCompactSpace G] [NoncompactSpace G]
    (μ : Measure G) [IsOpenPosMeasure μ] [μ.IsMulLeftInvariant] : μ univ = ∞

Proof state:
G : Type u_1
inst✝⁸ : MeasurableSpace G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : μ.IsMulLeftInvariant
⊢ μ univ = ⊤

Tactic:
obtain ⟨K, K1, hK, Kclosed⟩ : ∃ K ∈ 𝓝 (1 : G), IsCompact K ∧ IsClosed K :=
  exists_mem_nhds_isCompact_isClosed 1

Result:
case intro.intro.intro
G : Type u_1
inst✝⁸ : MeasurableSpace G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : μ.IsMulLeftInvariant
K : Set G
K1 : K ∈ 𝓝 1
hK : IsCompact K
Kclosed : IsClosed K
⊢ μ univ = ⊤

Proof state:
G : Type u_1
inst✝⁸ : MeasurableSpace G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : μ.IsMulLeftInvariant
K : Set G
K1 : K ∈ 𝓝 1
hK : IsCompact K
Kclosed : IsClosed K
K_pos : 0 < μ K
g : Set G → G
hg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)
L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
Lcompact : ∀ (n : ℕ), IsCompact (L n)
n : ℕ
⊢ IsClosed (L n)

Tactic:
induction' n with n IH

Result:
case zero
G : Type u_1
inst✝⁸ : MeasurableSpace G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : μ.IsMulLeftInvariant
K : Set G
K1 : K ∈ 𝓝 1
hK : IsCompact K
Kclosed : IsClosed K
K_pos : 0 < μ K
g : Set G → G
hg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)
L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
Lcompact : ∀ (n : ℕ), IsCompact (L n)
⊢ IsClosed (L 0)

case succ
G : Type u_1
inst✝⁸ : MeasurableSpace G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : μ.IsMulLeftInvariant
K : Set G
K1 : K ∈ 𝓝 1
hK : IsCompact K
Kclosed : IsClosed K
K_pos : 0 < μ K
g : Set G → G
hg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)
L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
Lcompact : ∀ (n : ℕ), IsCompact (L n)
n : ℕ
IH : IsClosed (L n)
⊢ IsClosed (L (n + 1))

Proof state:
case intro.intro.intro
G : Type u_1
inst✝⁸ : MeasurableSpace G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : μ.IsMulLeftInvariant
K : Set G
K1 : K ∈ 𝓝 1
hK : IsCompact K
Kclosed : IsClosed K
K_pos : 0 < μ K
g : Set G → G
hg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)
⊢ μ univ = ⊤

Tactic:
set L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K

Result:
case intro.intro.intro
G : Type u_1
inst✝⁸ : MeasurableSpace G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : μ.IsMulLeftInvariant
K : Set G
K1 : K ∈ 𝓝 1
hK : IsCompact K
Kclosed : IsClosed K
K_pos : 0 < μ K
g : Set G → G
hg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)
L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
⊢ μ univ = ⊤

Proof state:
G : Type u_1
inst✝⁸ : MeasurableSpace G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : μ.IsMulLeftInvariant
K : Set G
K1 : K ∈ 𝓝 1
hK : IsCompact K
Kclosed : IsClosed K
K_pos : 0 < μ K
g : Set G → G
hg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)
L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
n : ℕ
⊢ IsCompact (L n)

Tactic:
induction' n with n IH

Result:
case zero
G : Type u_1
inst✝⁸ : MeasurableSpace G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : μ.IsMulLeftInvariant
K : Set G
K1 : K ∈ 𝓝 1
hK : IsCompact K
Kclosed : IsClosed K
K_pos : 0 < μ K
g : Set G → G
hg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)
L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
⊢ IsCompact (L 0)

case succ
G : Type u_1
inst✝⁸ : MeasurableSpace G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : μ.IsMulLeftInvariant
K : Set G
K1 : K ∈ 𝓝 1
hK : IsCompact K
Kclosed : IsClosed K
K_pos : 0 < μ K
g : Set G → G
hg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)
L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
n : ℕ
IH : IsCompact (L n)
⊢ IsCompact (L (n + 1))

Proof state:
G : Type u_1
inst✝⁸ : MeasurableSpace G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : μ.IsMulLeftInvariant
K : Set G
K1 : K ∈ 𝓝 1
hK : IsCompact K
Kclosed : IsClosed K
K_pos : 0 < μ K
g : Set G → G
hg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)
L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
⊢ ∀ (n : ℕ), IsCompact (L n)

Tactic:
intro n

Result:
G : Type u_1
inst✝⁸ : MeasurableSpace G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : μ.IsMulLeftInvariant
K : Set G
K1 : K ∈ 𝓝 1
hK : IsCompact K
Kclosed : IsClosed K
K_pos : 0 < μ K
g : Set G → G
hg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)
L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
n : ℕ
⊢ IsCompact (L n)

Proof state:
case zero
G : Type u_1
inst✝⁸ : MeasurableSpace G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : μ.IsOpenPosMeasure
inst✝ : μ.IsMulLeftInvariant
K : Set G
K1 : K ∈ 𝓝 1
hK : IsCompact K
Kclosed : IsClosed K
K_pos : 0 < μ K
g : Set G → G
hg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)
L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
⊢ IsCompact (L 0)

Tactic:
exact hK

Result:
no goals