l3lab/ntp-mathlib huggingface dataset:
/-- In a noncompact locally compact group, a left-invariant measure which is positive
on open sets has infinite mass. -/
@[to_additive (attr := simp)
"In a noncompact locally compact additive group, a left-invariant measure which is positive on open
sets has infinite mass."]
theorem measure_univ_of_isMulLeftInvariant [WeaklyLocallyCompactSpace G] [NoncompactSpace G]
    (μ : Measure G) [IsOpenPosMeasure μ] [μ.IsMulLeftInvariant] : μ univ = ∞ := by
  /- Consider a closed compact set `K` with nonempty interior. For any compact set `L`, one may
    find `g = g (L)` such that `L` is disjoint from `g • K`. Iterating this, one finds
    infinitely many translates of `K` which are disjoint from each other. As they all have the
    same positive mass, it follows that the space has infinite measure. -/
  obtain ⟨K, hK, Kclosed, K1⟩ : ∃ K : Set G, IsCompact K ∧ IsClosed K ∧ K ∈ 𝓝 1 :=
    exists_isCompact_isClosed_nhds_one G
  have K_pos : 0 < μ K := measure_pos_of_nonempty_interior _ ⟨_, mem_interior_iff_mem_nhds.2 K1⟩
  have A : ∀ L : Set G, IsCompact L → ∃ g : G, Disjoint L (g • K) := fun L hL =>
    exists_disjoint_smul_of_isCompact hL hK
  choose! g hg using A
  

Proof state:
case intro.intro.intro
𝕜 : Type u_1
G : Type u_2
H : Type u_3
inst✝¹⁰ : MeasurableSpace G
inst✝⁹ : MeasurableSpace H
inst✝⁸ : TopologicalSpace G
inst✝⁷ : BorelSpace G
μ✝ : Measure G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : IsMulLeftInvariant μ✝
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : IsOpenPosMeasure μ
inst✝ : IsMulLeftInvariant μ
K : Set G
hK : IsCompact K
Kclosed : IsClosed K
K1 : K ∈ 𝓝 1
K_pos : 0 < ↑↑μ K
g : Set G → G
hg : ∀ (L : Set G), IsCompact L → Disjoint L (g L • K)
⊢ ↑↑μ univ = ⊤

Next tactic:
set L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
/-- In a noncompact locally compact group, a left-invariant measure which is positive
on open sets has infinite mass. -/
@[to_additive (attr := simp)
"In a noncompact locally compact additive group, a left-invariant measure which is positive on open
sets has infinite mass."]
theorem measure_univ_of_isMulLeftInvariant [WeaklyLocallyCompactSpace G] [NoncompactSpace G]
    (μ : Measure G) [IsOpenPosMeasure μ] [μ.IsMulLeftInvariant] : μ univ = ∞ := by
  /- Consider a closed compact set `K` with nonempty interior. For any compact set `L`, one may
    find `g = g (L)` such that `L` is disjoint from `g • K`. Iterating this, one finds
    infinitely many translates of `K` which are disjoint from each other. As they all have the
    same positive mass, it follows that the space has infinite measure. -/
  obtain ⟨K, hK, Kclosed, K1⟩ : ∃ K : Set G, IsCompact K ∧ IsClosed K ∧ K ∈ 𝓝 1 :=
    exists_isCompact_isClosed_nhds_one G
  have K_pos : 0 < μ K := measure_pos_of_nonempty_interior _ ⟨_, mem_interior_iff_mem_nhds.2 K1⟩
  have A : ∀ L : Set G, IsCompact L → ∃ g : G, Disjoint L (g • K) := fun L hL =>
    exists_disjoint_smul_of_isCompact hL hK
  choose! g hg using A
  set L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
  

Proof state:
case intro.intro.intro
𝕜 : Type u_1
G : Type u_2
H : Type u_3
inst✝¹⁰ : MeasurableSpace G
inst✝⁹ : MeasurableSpace H
inst✝⁸ : TopologicalSpace G
inst✝⁷ : BorelSpace G
μ✝ : Measure G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : IsMulLeftInvariant μ✝
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : IsOpenPosMeasure μ
inst✝ : IsMulLeftInvariant μ
K : Set G
hK : IsCompact K
Kclosed : IsClosed K
K1 : K ∈ 𝓝 1
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 = ⊤

Next tactic:
have Lcompact : ∀ n, IsCompact (L n) := by
    intro n
    induction' n with n IH
    · exact hK
    · simp_rw [iterate_succ']
      apply IsCompact.union IH (hK.smul (g (L n)))
/-- In a noncompact locally compact group, a left-invariant measure which is positive
on open sets has infinite mass. -/
@[to_additive (attr := simp)
"In a noncompact locally compact additive group, a left-invariant measure which is positive on open
sets has infinite mass."]
theorem measure_univ_of_isMulLeftInvariant [WeaklyLocallyCompactSpace G] [NoncompactSpace G]
    (μ : Measure G) [IsOpenPosMeasure μ] [μ.IsMulLeftInvariant] : μ univ = ∞ := by
  /- Consider a closed compact set `K` with nonempty interior. For any compact set `L`, one may
    find `g = g (L)` such that `L` is disjoint from `g • K`. Iterating this, one finds
    infinitely many translates of `K` which are disjoint from each other. As they all have the
    same positive mass, it follows that the space has infinite measure. -/
  obtain ⟨K, hK, Kclosed, K1⟩ : ∃ K : Set G, IsCompact K ∧ IsClosed K ∧ K ∈ 𝓝 1 :=
    exists_isCompact_isClosed_nhds_one G
  have K_pos : 0 < μ K := measure_pos_of_nonempty_interior _ ⟨_, mem_interior_iff_mem_nhds.2 K1⟩
  have A : ∀ L : Set G, IsCompact L → ∃ g : G, Disjoint L (g • K) := fun L hL =>
    exists_disjoint_smul_of_isCompact hL hK
  choose! g hg using A
  set L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
  have Lcompact : ∀ n, IsCompact (L n) := by
    

Proof state:
𝕜 : Type u_1
G : Type u_2
H : Type u_3
inst✝¹⁰ : MeasurableSpace G
inst✝⁹ : MeasurableSpace H
inst✝⁸ : TopologicalSpace G
inst✝⁷ : BorelSpace G
μ✝ : Measure G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : IsMulLeftInvariant μ✝
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : IsOpenPosMeasure μ
inst✝ : IsMulLeftInvariant μ
K : Set G
hK : IsCompact K
Kclosed : IsClosed K
K1 : K ∈ 𝓝 1
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)

Next tactic:
intro n
/-- In a noncompact locally compact group, a left-invariant measure which is positive
on open sets has infinite mass. -/
@[to_additive (attr := simp)
"In a noncompact locally compact additive group, a left-invariant measure which is positive on open
sets has infinite mass."]
theorem measure_univ_of_isMulLeftInvariant [WeaklyLocallyCompactSpace G] [NoncompactSpace G]
    (μ : Measure G) [IsOpenPosMeasure μ] [μ.IsMulLeftInvariant] : μ univ = ∞ := by
  /- Consider a closed compact set `K` with nonempty interior. For any compact set `L`, one may
    find `g = g (L)` such that `L` is disjoint from `g • K`. Iterating this, one finds
    infinitely many translates of `K` which are disjoint from each other. As they all have the
    same positive mass, it follows that the space has infinite measure. -/
  obtain ⟨K, hK, Kclosed, K1⟩ : ∃ K : Set G, IsCompact K ∧ IsClosed K ∧ K ∈ 𝓝 1 :=
    exists_isCompact_isClosed_nhds_one G
  have K_pos : 0 < μ K := measure_pos_of_nonempty_interior _ ⟨_, mem_interior_iff_mem_nhds.2 K1⟩
  have A : ∀ L : Set G, IsCompact L → ∃ g : G, Disjoint L (g • K) := fun L hL =>
    exists_disjoint_smul_of_isCompact hL hK
  choose! g hg using A
  set L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
  have Lcompact : ∀ n, IsCompact (L n) := by
    intro n
    

Proof state:
𝕜 : Type u_1
G : Type u_2
H : Type u_3
inst✝¹⁰ : MeasurableSpace G
inst✝⁹ : MeasurableSpace H
inst✝⁸ : TopologicalSpace G
inst✝⁷ : BorelSpace G
μ✝ : Measure G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : IsMulLeftInvariant μ✝
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : IsOpenPosMeasure μ
inst✝ : IsMulLeftInvariant μ
K : Set G
hK : IsCompact K
Kclosed : IsClosed K
K1 : K ∈ 𝓝 1
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)

Next tactic:
induction' n with n IH
/-- In a noncompact locally compact group, a left-invariant measure which is positive
on open sets has infinite mass. -/
@[to_additive (attr := simp)
"In a noncompact locally compact additive group, a left-invariant measure which is positive on open
sets has infinite mass."]
theorem measure_univ_of_isMulLeftInvariant [WeaklyLocallyCompactSpace G] [NoncompactSpace G]
    (μ : Measure G) [IsOpenPosMeasure μ] [μ.IsMulLeftInvariant] : μ univ = ∞ := by
  /- Consider a closed compact set `K` with nonempty interior. For any compact set `L`, one may
    find `g = g (L)` such that `L` is disjoint from `g • K`. Iterating this, one finds
    infinitely many translates of `K` which are disjoint from each other. As they all have the
    same positive mass, it follows that the space has infinite measure. -/
  obtain ⟨K, hK, Kclosed, K1⟩ : ∃ K : Set G, IsCompact K ∧ IsClosed K ∧ K ∈ 𝓝 1 :=
    exists_isCompact_isClosed_nhds_one G
  have K_pos : 0 < μ K := measure_pos_of_nonempty_interior _ ⟨_, mem_interior_iff_mem_nhds.2 K1⟩
  have A : ∀ L : Set G, IsCompact L → ∃ g : G, Disjoint L (g • K) := fun L hL =>
    exists_disjoint_smul_of_isCompact hL hK
  choose! g hg using A
  set L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
  have Lcompact : ∀ n, IsCompact (L n) := by
    intro n
    induction' n with n IH
    · 

Proof state:
case zero
𝕜 : Type u_1
G : Type u_2
H : Type u_3
inst✝¹⁰ : MeasurableSpace G
inst✝⁹ : MeasurableSpace H
inst✝⁸ : TopologicalSpace G
inst✝⁷ : BorelSpace G
μ✝ : Measure G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : IsMulLeftInvariant μ✝
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : IsOpenPosMeasure μ
inst✝ : IsMulLeftInvariant μ
K : Set G
hK : IsCompact K
Kclosed : IsClosed K
K1 : K ∈ 𝓝 1
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 Nat.zero)

Next tactic:
exact hK
/-- In a noncompact locally compact group, a left-invariant measure which is positive
on open sets has infinite mass. -/
@[to_additive (attr := simp)
"In a noncompact locally compact additive group, a left-invariant measure which is positive on open
sets has infinite mass."]
theorem measure_univ_of_isMulLeftInvariant [WeaklyLocallyCompactSpace G] [NoncompactSpace G]
    (μ : Measure G) [IsOpenPosMeasure μ] [μ.IsMulLeftInvariant] : μ univ = ∞ := by
  /- Consider a closed compact set `K` with nonempty interior. For any compact set `L`, one may
    find `g = g (L)` such that `L` is disjoint from `g • K`. Iterating this, one finds
    infinitely many translates of `K` which are disjoint from each other. As they all have the
    same positive mass, it follows that the space has infinite measure. -/
  obtain ⟨K, hK, Kclosed, K1⟩ : ∃ K : Set G, IsCompact K ∧ IsClosed K ∧ K ∈ 𝓝 1 :=
    exists_isCompact_isClosed_nhds_one G
  have K_pos : 0 < μ K := measure_pos_of_nonempty_interior _ ⟨_, mem_interior_iff_mem_nhds.2 K1⟩
  have A : ∀ L : Set G, IsCompact L → ∃ g : G, Disjoint L (g • K) := fun L hL =>
    exists_disjoint_smul_of_isCompact hL hK
  choose! g hg using A
  set L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
  have Lcompact : ∀ n, IsCompact (L n) := by
    intro n
    induction' n with n IH
    · exact hK
    · 

Proof state:
case succ
𝕜 : Type u_1
G : Type u_2
H : Type u_3
inst✝¹⁰ : MeasurableSpace G
inst✝⁹ : MeasurableSpace H
inst✝⁸ : TopologicalSpace G
inst✝⁷ : BorelSpace G
μ✝ : Measure G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : IsMulLeftInvariant μ✝
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : IsOpenPosMeasure μ
inst✝ : IsMulLeftInvariant μ
K : Set G
hK : IsCompact K
Kclosed : IsClosed K
K1 : K ∈ 𝓝 1
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 (Nat.succ n))

Next tactic:
simp_rw [iterate_succ']
/-- In a noncompact locally compact group, a left-invariant measure which is positive
on open sets has infinite mass. -/
@[to_additive (attr := simp)
"In a noncompact locally compact additive group, a left-invariant measure which is positive on open
sets has infinite mass."]
theorem measure_univ_of_isMulLeftInvariant [WeaklyLocallyCompactSpace G] [NoncompactSpace G]
    (μ : Measure G) [IsOpenPosMeasure μ] [μ.IsMulLeftInvariant] : μ univ = ∞ := by
  /- Consider a closed compact set `K` with nonempty interior. For any compact set `L`, one may
    find `g = g (L)` such that `L` is disjoint from `g • K`. Iterating this, one finds
    infinitely many translates of `K` which are disjoint from each other. As they all have the
    same positive mass, it follows that the space has infinite measure. -/
  obtain ⟨K, hK, Kclosed, K1⟩ : ∃ K : Set G, IsCompact K ∧ IsClosed K ∧ K ∈ 𝓝 1 :=
    exists_isCompact_isClosed_nhds_one G
  have K_pos : 0 < μ K := measure_pos_of_nonempty_interior _ ⟨_, mem_interior_iff_mem_nhds.2 K1⟩
  have A : ∀ L : Set G, IsCompact L → ∃ g : G, Disjoint L (g • K) := fun L hL =>
    exists_disjoint_smul_of_isCompact hL hK
  choose! g hg using A
  set L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
  have Lcompact : ∀ n, IsCompact (L n) := by
    intro n
    induction' n with n IH
    · exact hK
    · simp_rw [iterate_succ']
      

Proof state:
case succ
𝕜 : Type u_1
G : Type u_2
H : Type u_3
inst✝¹⁰ : MeasurableSpace G
inst✝⁹ : MeasurableSpace H
inst✝⁸ : TopologicalSpace G
inst✝⁷ : BorelSpace G
μ✝ : Measure G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : IsMulLeftInvariant μ✝
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : IsOpenPosMeasure μ
inst✝ : IsMulLeftInvariant μ
K : Set G
hK : IsCompact K
Kclosed : IsClosed K
K1 : K ∈ 𝓝 1
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)

Next tactic:
apply IsCompact.union IH (hK.smul (g (L n)))
/-- In a noncompact locally compact group, a left-invariant measure which is positive
on open sets has infinite mass. -/
@[to_additive (attr := simp)
"In a noncompact locally compact additive group, a left-invariant measure which is positive on open
sets has infinite mass."]
theorem measure_univ_of_isMulLeftInvariant [WeaklyLocallyCompactSpace G] [NoncompactSpace G]
    (μ : Measure G) [IsOpenPosMeasure μ] [μ.IsMulLeftInvariant] : μ univ = ∞ := by
  /- Consider a closed compact set `K` with nonempty interior. For any compact set `L`, one may
    find `g = g (L)` such that `L` is disjoint from `g • K`. Iterating this, one finds
    infinitely many translates of `K` which are disjoint from each other. As they all have the
    same positive mass, it follows that the space has infinite measure. -/
  obtain ⟨K, hK, Kclosed, K1⟩ : ∃ K : Set G, IsCompact K ∧ IsClosed K ∧ K ∈ 𝓝 1 :=
    exists_isCompact_isClosed_nhds_one G
  have K_pos : 0 < μ K := measure_pos_of_nonempty_interior _ ⟨_, mem_interior_iff_mem_nhds.2 K1⟩
  have A : ∀ L : Set G, IsCompact L → ∃ g : G, Disjoint L (g • K) := fun L hL =>
    exists_disjoint_smul_of_isCompact hL hK
  choose! g hg using A
  set L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
  have Lcompact : ∀ n, IsCompact (L n) := by
    intro n
    induction' n with n IH
    · exact hK
    · simp_rw [iterate_succ']
      apply IsCompact.union IH (hK.smul (g (L n)))
  

Proof state:
case intro.intro.intro
𝕜 : Type u_1
G : Type u_2
H : Type u_3
inst✝¹⁰ : MeasurableSpace G
inst✝⁹ : MeasurableSpace H
inst✝⁸ : TopologicalSpace G
inst✝⁷ : BorelSpace G
μ✝ : Measure G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : IsMulLeftInvariant μ✝
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : IsOpenPosMeasure μ
inst✝ : IsMulLeftInvariant μ
K : Set G
hK : IsCompact K
Kclosed : IsClosed K
K1 : K ∈ 𝓝 1
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)
⊢ ↑↑μ univ = ⊤

Next tactic:
have Lclosed : ∀ n, IsClosed (L n) := by
    intro n
    induction' n with n IH
    · exact Kclosed
    · simp_rw [iterate_succ']
      apply IsClosed.union IH (Kclosed.smul (g (L n)))
/-- In a noncompact locally compact group, a left-invariant measure which is positive
on open sets has infinite mass. -/
@[to_additive (attr := simp)
"In a noncompact locally compact additive group, a left-invariant measure which is positive on open
sets has infinite mass."]
theorem measure_univ_of_isMulLeftInvariant [WeaklyLocallyCompactSpace G] [NoncompactSpace G]
    (μ : Measure G) [IsOpenPosMeasure μ] [μ.IsMulLeftInvariant] : μ univ = ∞ := by
  /- Consider a closed compact set `K` with nonempty interior. For any compact set `L`, one may
    find `g = g (L)` such that `L` is disjoint from `g • K`. Iterating this, one finds
    infinitely many translates of `K` which are disjoint from each other. As they all have the
    same positive mass, it follows that the space has infinite measure. -/
  obtain ⟨K, hK, Kclosed, K1⟩ : ∃ K : Set G, IsCompact K ∧ IsClosed K ∧ K ∈ 𝓝 1 :=
    exists_isCompact_isClosed_nhds_one G
  have K_pos : 0 < μ K := measure_pos_of_nonempty_interior _ ⟨_, mem_interior_iff_mem_nhds.2 K1⟩
  have A : ∀ L : Set G, IsCompact L → ∃ g : G, Disjoint L (g • K) := fun L hL =>
    exists_disjoint_smul_of_isCompact hL hK
  choose! g hg using A
  set L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
  have Lcompact : ∀ n, IsCompact (L n) := by
    intro n
    induction' n with n IH
    · exact hK
    · simp_rw [iterate_succ']
      apply IsCompact.union IH (hK.smul (g (L n)))
  have Lclosed : ∀ n, IsClosed (L n) := by
    

Proof state:
𝕜 : Type u_1
G : Type u_2
H : Type u_3
inst✝¹⁰ : MeasurableSpace G
inst✝⁹ : MeasurableSpace H
inst✝⁸ : TopologicalSpace G
inst✝⁷ : BorelSpace G
μ✝ : Measure G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : IsMulLeftInvariant μ✝
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : IsOpenPosMeasure μ
inst✝ : IsMulLeftInvariant μ
K : Set G
hK : IsCompact K
Kclosed : IsClosed K
K1 : K ∈ 𝓝 1
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)

Next tactic:
intro n
/-- In a noncompact locally compact group, a left-invariant measure which is positive
on open sets has infinite mass. -/
@[to_additive (attr := simp)
"In a noncompact locally compact additive group, a left-invariant measure which is positive on open
sets has infinite mass."]
theorem measure_univ_of_isMulLeftInvariant [WeaklyLocallyCompactSpace G] [NoncompactSpace G]
    (μ : Measure G) [IsOpenPosMeasure μ] [μ.IsMulLeftInvariant] : μ univ = ∞ := by
  /- Consider a closed compact set `K` with nonempty interior. For any compact set `L`, one may
    find `g = g (L)` such that `L` is disjoint from `g • K`. Iterating this, one finds
    infinitely many translates of `K` which are disjoint from each other. As they all have the
    same positive mass, it follows that the space has infinite measure. -/
  obtain ⟨K, hK, Kclosed, K1⟩ : ∃ K : Set G, IsCompact K ∧ IsClosed K ∧ K ∈ 𝓝 1 :=
    exists_isCompact_isClosed_nhds_one G
  have K_pos : 0 < μ K := measure_pos_of_nonempty_interior _ ⟨_, mem_interior_iff_mem_nhds.2 K1⟩
  have A : ∀ L : Set G, IsCompact L → ∃ g : G, Disjoint L (g • K) := fun L hL =>
    exists_disjoint_smul_of_isCompact hL hK
  choose! g hg using A
  set L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
  have Lcompact : ∀ n, IsCompact (L n) := by
    intro n
    induction' n with n IH
    · exact hK
    · simp_rw [iterate_succ']
      apply IsCompact.union IH (hK.smul (g (L n)))
  have Lclosed : ∀ n, IsClosed (L n) := by
    intro n
    

Proof state:
𝕜 : Type u_1
G : Type u_2
H : Type u_3
inst✝¹⁰ : MeasurableSpace G
inst✝⁹ : MeasurableSpace H
inst✝⁸ : TopologicalSpace G
inst✝⁷ : BorelSpace G
μ✝ : Measure G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : IsMulLeftInvariant μ✝
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : IsOpenPosMeasure μ
inst✝ : IsMulLeftInvariant μ
K : Set G
hK : IsCompact K
Kclosed : IsClosed K
K1 : K ∈ 𝓝 1
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)

Next tactic:
induction' n with n IH
/-- In a noncompact locally compact group, a left-invariant measure which is positive
on open sets has infinite mass. -/
@[to_additive (attr := simp)
"In a noncompact locally compact additive group, a left-invariant measure which is positive on open
sets has infinite mass."]
theorem measure_univ_of_isMulLeftInvariant [WeaklyLocallyCompactSpace G] [NoncompactSpace G]
    (μ : Measure G) [IsOpenPosMeasure μ] [μ.IsMulLeftInvariant] : μ univ = ∞ := by
  /- Consider a closed compact set `K` with nonempty interior. For any compact set `L`, one may
    find `g = g (L)` such that `L` is disjoint from `g • K`. Iterating this, one finds
    infinitely many translates of `K` which are disjoint from each other. As they all have the
    same positive mass, it follows that the space has infinite measure. -/
  obtain ⟨K, hK, Kclosed, K1⟩ : ∃ K : Set G, IsCompact K ∧ IsClosed K ∧ K ∈ 𝓝 1 :=
    exists_isCompact_isClosed_nhds_one G
  have K_pos : 0 < μ K := measure_pos_of_nonempty_interior _ ⟨_, mem_interior_iff_mem_nhds.2 K1⟩
  have A : ∀ L : Set G, IsCompact L → ∃ g : G, Disjoint L (g • K) := fun L hL =>
    exists_disjoint_smul_of_isCompact hL hK
  choose! g hg using A
  set L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
  have Lcompact : ∀ n, IsCompact (L n) := by
    intro n
    induction' n with n IH
    · exact hK
    · simp_rw [iterate_succ']
      apply IsCompact.union IH (hK.smul (g (L n)))
  have Lclosed : ∀ n, IsClosed (L n) := by
    intro n
    induction' n with n IH
    · 

Proof state:
case zero
𝕜 : Type u_1
G : Type u_2
H : Type u_3
inst✝¹⁰ : MeasurableSpace G
inst✝⁹ : MeasurableSpace H
inst✝⁸ : TopologicalSpace G
inst✝⁷ : BorelSpace G
μ✝ : Measure G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : IsMulLeftInvariant μ✝
inst✝³ : WeaklyLocallyCompactSpace G
inst✝² : NoncompactSpace G
μ : Measure G
inst✝¹ : IsOpenPosMeasure μ
inst✝ : IsMulLeftInvariant μ
K : Set G
hK : IsCompact K
Kclosed : IsClosed K
K1 : K ∈ 𝓝 1
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 Nat.zero)

Next tactic:
exact Kclosed