-- 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:
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
Lcompact : ∀ (n : ℕ), IsCompact (L n)
Lclosed : ∀ (n : ℕ), IsClosed (L n)
M : ∀ (n : ℕ), μ (L n) = ↑(n + 1) * μ K
N : Tendsto (fun n => μ (L n)) atTop (𝓝 (⊤ * μ K))
⊢ μ univ = ⊤

Tactic:
simp only [ENNReal.top_mul', K_pos.ne', if_false] at N

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
Lcompact : ∀ (n : ℕ), IsCompact (L n)
Lclosed : ∀ (n : ℕ), IsClosed (L n)
M : ∀ (n : ℕ), μ (L n) = ↑(n + 1) * μ K
N : Tendsto (fun n => μ (L n)) atTop (𝓝 ⊤)
⊢ 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:
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
Lcompact : ∀ (n : ℕ), IsCompact (L n)
n : ℕ
⊢ IsClosed (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
Lcompact : ∀ (n : ℕ), IsCompact (L n)
⊢ IsClosed (L 0)

Tactic:
exact Kclosed

Result:
no goals

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)
Lclosed : ∀ (n : ℕ), IsClosed (L n)
n : ℕ
⊢ μ (L n) = ↑(n + 1) * μ K

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)
Lclosed : ∀ (n : ℕ), IsClosed (L n)
⊢ μ (L 0) = ↑(0 + 1) * μ K

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)
Lclosed : ∀ (n : ℕ), IsClosed (L n)
n : ℕ
IH : μ (L n) = ↑(n + 1) * μ K
⊢ μ (L (n + 1)) = ↑(n + 1 + 1) * μ K

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
⊢ μ univ = ⊤

Tactic:
have A : ∀ L : Set G, IsCompact L → ∃ g : G, Disjoint L (g • K) := fun L hL =>
  exists_disjoint_smul_of_isCompact hL hK

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
A : ∀ (L : Set G), IsCompact L → ∃ g, Disjoint L (g • K)
⊢ μ univ = ⊤

Proof state:
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 (((fun T => T ∪ g T • K) ∘ (fun T => T ∪ g T • K)^[n]) K)

Tactic:
apply IsCompact.union IH (hK.smul (g (L n)))

Result:
no goals