l3lab/ntp-mathlib huggingface dataset:
/-- If a set has zero measure in a neighborhood of each of its points, then it has zero measure
in a second-countable space. -/
theorem null_of_locally_null [TopologicalSpace α] [SecondCountableTopology α] (m : OuterMeasure α)
    (s : Set α) (hs : ∀ x ∈ s, ∃ u ∈ 𝓝[s] x, m u = 0) : m s = 0 := by
  

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ : OuterMeasure α
inst✝¹ : TopologicalSpace α
inst✝ : SecondCountableTopology α
m : OuterMeasure α
s : Set α
hs : ∀ x ∈ s, ∃ u ∈ 𝓝[s] x, ↑m u = 0
⊢ ↑m s = 0

Next tactic:
choose! u hxu hu₀ using hs
/-- If a set has zero measure in a neighborhood of each of its points, then it has zero measure
in a second-countable space. -/
theorem null_of_locally_null [TopologicalSpace α] [SecondCountableTopology α] (m : OuterMeasure α)
    (s : Set α) (hs : ∀ x ∈ s, ∃ u ∈ 𝓝[s] x, m u = 0) : m s = 0 := by
  choose! u hxu hu₀ using hs
  

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ : OuterMeasure α
inst✝¹ : TopologicalSpace α
inst✝ : SecondCountableTopology α
m : OuterMeasure α
s : Set α
u : α → Set α
hxu : ∀ x ∈ s, u x ∈ 𝓝[s] x
hu₀ : ∀ x ∈ s, ↑m (u x) = 0
⊢ ↑m s = 0

Next tactic:
choose t ht using TopologicalSpace.countable_cover_nhdsWithin hxu
/-- If a set has zero measure in a neighborhood of each of its points, then it has zero measure
in a second-countable space. -/
theorem null_of_locally_null [TopologicalSpace α] [SecondCountableTopology α] (m : OuterMeasure α)
    (s : Set α) (hs : ∀ x ∈ s, ∃ u ∈ 𝓝[s] x, m u = 0) : m s = 0 := by
  choose! u hxu hu₀ using hs
  choose t ht using TopologicalSpace.countable_cover_nhdsWithin hxu
  

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ : OuterMeasure α
inst✝¹ : TopologicalSpace α
inst✝ : SecondCountableTopology α
m : OuterMeasure α
s : Set α
u : α → Set α
hxu : ∀ x ∈ s, u x ∈ 𝓝[s] x
hu₀ : ∀ x ∈ s, ↑m (u x) = 0
t : Set α
ht : t ⊆ s ∧ Set.Countable t ∧ s ⊆ ⋃ x ∈ t, u x
⊢ ↑m s = 0

Next tactic:
rcases ht with ⟨ts, t_count, ht⟩
/-- If a set has zero measure in a neighborhood of each of its points, then it has zero measure
in a second-countable space. -/
theorem null_of_locally_null [TopologicalSpace α] [SecondCountableTopology α] (m : OuterMeasure α)
    (s : Set α) (hs : ∀ x ∈ s, ∃ u ∈ 𝓝[s] x, m u = 0) : m s = 0 := by
  choose! u hxu hu₀ using hs
  choose t ht using TopologicalSpace.countable_cover_nhdsWithin hxu
  rcases ht with ⟨ts, t_count, ht⟩
  

Proof state:
case intro.intro
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ : OuterMeasure α
inst✝¹ : TopologicalSpace α
inst✝ : SecondCountableTopology α
m : OuterMeasure α
s : Set α
u : α → Set α
hxu : ∀ x ∈ s, u x ∈ 𝓝[s] x
hu₀ : ∀ x ∈ s, ↑m (u x) = 0
t : Set α
ts : t ⊆ s
t_count : Set.Countable t
ht : s ⊆ ⋃ x ∈ t, u x
⊢ ↑m s = 0

Next tactic:
apply m.mono_null ht
/-- If a set has zero measure in a neighborhood of each of its points, then it has zero measure
in a second-countable space. -/
theorem null_of_locally_null [TopologicalSpace α] [SecondCountableTopology α] (m : OuterMeasure α)
    (s : Set α) (hs : ∀ x ∈ s, ∃ u ∈ 𝓝[s] x, m u = 0) : m s = 0 := by
  choose! u hxu hu₀ using hs
  choose t ht using TopologicalSpace.countable_cover_nhdsWithin hxu
  rcases ht with ⟨ts, t_count, ht⟩
  apply m.mono_null ht
  

Proof state:
case intro.intro
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ : OuterMeasure α
inst✝¹ : TopologicalSpace α
inst✝ : SecondCountableTopology α
m : OuterMeasure α
s : Set α
u : α → Set α
hxu : ∀ x ∈ s, u x ∈ 𝓝[s] x
hu₀ : ∀ x ∈ s, ↑m (u x) = 0
t : Set α
ts : t ⊆ s
t_count : Set.Countable t
ht : s ⊆ ⋃ x ∈ t, u x
⊢ ↑m (⋃ x ∈ t, u x) = 0

Next tactic:
exact (m.biUnion_null_iff t_count).2 fun x hx => hu₀ x (ts hx)
/-- If `m s ≠ 0`, then for some point `x ∈ s` and any `t ∈ 𝓝[s] x` we have `0 < m t`. -/
theorem exists_mem_forall_mem_nhds_within_pos [TopologicalSpace α] [SecondCountableTopology α]
    (m : OuterMeasure α) {s : Set α} (hs : m s ≠ 0) : ∃ x ∈ s, ∀ t ∈ 𝓝[s] x, 0 < m t := by
  

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ : OuterMeasure α
inst✝¹ : TopologicalSpace α
inst✝ : SecondCountableTopology α
m : OuterMeasure α
s : Set α
hs : ↑m s ≠ 0
⊢ ∃ x ∈ s, ∀ t ∈ 𝓝[s] x, 0 < ↑m t

Next tactic:
contrapose! hs
/-- If `m s ≠ 0`, then for some point `x ∈ s` and any `t ∈ 𝓝[s] x` we have `0 < m t`. -/
theorem exists_mem_forall_mem_nhds_within_pos [TopologicalSpace α] [SecondCountableTopology α]
    (m : OuterMeasure α) {s : Set α} (hs : m s ≠ 0) : ∃ x ∈ s, ∀ t ∈ 𝓝[s] x, 0 < m t := by
  contrapose! hs
  

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ : OuterMeasure α
inst✝¹ : TopologicalSpace α
inst✝ : SecondCountableTopology α
m : OuterMeasure α
s : Set α
hs : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, ↑m t ≤ 0
⊢ ↑m s = 0

Next tactic:
simp only [nonpos_iff_eq_zero] at hs
/-- If `m s ≠ 0`, then for some point `x ∈ s` and any `t ∈ 𝓝[s] x` we have `0 < m t`. -/
theorem exists_mem_forall_mem_nhds_within_pos [TopologicalSpace α] [SecondCountableTopology α]
    (m : OuterMeasure α) {s : Set α} (hs : m s ≠ 0) : ∃ x ∈ s, ∀ t ∈ 𝓝[s] x, 0 < m t := by
  contrapose! hs
  simp only [nonpos_iff_eq_zero] at hs
  

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ : OuterMeasure α
inst✝¹ : TopologicalSpace α
inst✝ : SecondCountableTopology α
m : OuterMeasure α
s : Set α
hs : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, ↑m t = 0
⊢ ↑m s = 0

Next tactic:
exact m.null_of_locally_null s hs
/-- If `s : ι → Set α` is a sequence of sets, `S = ⋃ n, s n`, and `m (S \ s n)` tends to zero along
some nontrivial filter (usually `atTop` on `ι = ℕ`), then `m S = ⨆ n, m (s n)`. -/
theorem iUnion_of_tendsto_zero {ι} (m : OuterMeasure α) {s : ι → Set α} (l : Filter ι) [NeBot l]
    (h0 : Tendsto (fun k => m ((⋃ n, s n) \ s k)) l (𝓝 0)) : m (⋃ n, s n) = ⨆ n, m (s n) := by
  

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ : OuterMeasure α
ι : Type u_5
m : OuterMeasure α
s : ι → Set α
l : Filter ι
inst✝ : NeBot l
h0 : Tendsto (fun k => ↑m ((⋃ n, s n) \ s k)) l (𝓝 0)
⊢ ↑m (⋃ n, s n) = ⨆ n, ↑m (s n)

Next tactic:
set S := ⋃ n, s n
/-- If `s : ι → Set α` is a sequence of sets, `S = ⋃ n, s n`, and `m (S \ s n)` tends to zero along
some nontrivial filter (usually `atTop` on `ι = ℕ`), then `m S = ⨆ n, m (s n)`. -/
theorem iUnion_of_tendsto_zero {ι} (m : OuterMeasure α) {s : ι → Set α} (l : Filter ι) [NeBot l]
    (h0 : Tendsto (fun k => m ((⋃ n, s n) \ s k)) l (𝓝 0)) : m (⋃ n, s n) = ⨆ n, m (s n) := by
  set S := ⋃ n, s n
  

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ : OuterMeasure α
ι : Type u_5
m : OuterMeasure α
s : ι → Set α
l : Filter ι
inst✝ : NeBot l
S : Set α := ⋃ n, s n
h0 : Tendsto (fun k => ↑m (S \ s k)) l (𝓝 0)
⊢ ↑m S = ⨆ n, ↑m (s n)

Next tactic:
set M := ⨆ n, m (s n)
/-- If `s : ι → Set α` is a sequence of sets, `S = ⋃ n, s n`, and `m (S \ s n)` tends to zero along
some nontrivial filter (usually `atTop` on `ι = ℕ`), then `m S = ⨆ n, m (s n)`. -/
theorem iUnion_of_tendsto_zero {ι} (m : OuterMeasure α) {s : ι → Set α} (l : Filter ι) [NeBot l]
    (h0 : Tendsto (fun k => m ((⋃ n, s n) \ s k)) l (𝓝 0)) : m (⋃ n, s n) = ⨆ n, m (s n) := by
  set S := ⋃ n, s n
  set M := ⨆ n, m (s n)
  

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ : OuterMeasure α
ι : Type u_5
m : OuterMeasure α
s : ι → Set α
l : Filter ι
inst✝ : NeBot l
S : Set α := ⋃ n, s n
h0 : Tendsto (fun k => ↑m (S \ s k)) l (𝓝 0)
M : ℝ≥0∞ := ⨆ n, ↑m (s n)
⊢ ↑m S = M

Next tactic:
have hsS : ∀ {k}, s k ⊆ S := fun {k} => subset_iUnion _ _
/-- If `s : ι → Set α` is a sequence of sets, `S = ⋃ n, s n`, and `m (S \ s n)` tends to zero along
some nontrivial filter (usually `atTop` on `ι = ℕ`), then `m S = ⨆ n, m (s n)`. -/
theorem iUnion_of_tendsto_zero {ι} (m : OuterMeasure α) {s : ι → Set α} (l : Filter ι) [NeBot l]
    (h0 : Tendsto (fun k => m ((⋃ n, s n) \ s k)) l (𝓝 0)) : m (⋃ n, s n) = ⨆ n, m (s n) := by
  set S := ⋃ n, s n
  set M := ⨆ n, m (s n)
  have hsS : ∀ {k}, s k ⊆ S := fun {k} => subset_iUnion _ _
  

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ : OuterMeasure α
ι : Type u_5
m : OuterMeasure α
s : ι → Set α
l : Filter ι
inst✝ : NeBot l
S : Set α := ⋃ n, s n
h0 : Tendsto (fun k => ↑m (S \ s k)) l (𝓝 0)
M : ℝ≥0∞ := ⨆ n, ↑m (s n)
hsS : ∀ {k : ι}, s k ⊆ S
⊢ ↑m S = M

Next tactic:
refine' le_antisymm _ (iSup_le fun n => m.mono hsS)
/-- If `s : ι → Set α` is a sequence of sets, `S = ⋃ n, s n`, and `m (S \ s n)` tends to zero along
some nontrivial filter (usually `atTop` on `ι = ℕ`), then `m S = ⨆ n, m (s n)`. -/
theorem iUnion_of_tendsto_zero {ι} (m : OuterMeasure α) {s : ι → Set α} (l : Filter ι) [NeBot l]
    (h0 : Tendsto (fun k => m ((⋃ n, s n) \ s k)) l (𝓝 0)) : m (⋃ n, s n) = ⨆ n, m (s n) := by
  set S := ⋃ n, s n
  set M := ⨆ n, m (s n)
  have hsS : ∀ {k}, s k ⊆ S := fun {k} => subset_iUnion _ _
  refine' le_antisymm _ (iSup_le fun n => m.mono hsS)
  

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ : OuterMeasure α
ι : Type u_5
m : OuterMeasure α
s : ι → Set α
l : Filter ι
inst✝ : NeBot l
S : Set α := ⋃ n, s n
h0 : Tendsto (fun k => ↑m (S \ s k)) l (𝓝 0)
M : ℝ≥0∞ := ⨆ n, ↑m (s n)
hsS : ∀ {k : ι}, s k ⊆ S
⊢ ↑m S ≤ M

Next tactic:
have A : ∀ k, m S ≤ M + m (S \ s k) := fun k =>
    calc
      m S = m (s k ∪ S \ s k) := by rw [union_diff_self, union_eq_self_of_subset_left hsS]
      _ ≤ m (s k) + m (S \ s k) := (m.union _ _)
      _ ≤ M + m (S \ s k) := add_le_add_right (le_iSup (m.measureOf ∘ s) k) _
/-- If `s : ι → Set α` is a sequence of sets, `S = ⋃ n, s n`, and `m (S \ s n)` tends to zero along
some nontrivial filter (usually `atTop` on `ι = ℕ`), then `m S = ⨆ n, m (s n)`. -/
theorem iUnion_of_tendsto_zero {ι} (m : OuterMeasure α) {s : ι → Set α} (l : Filter ι) [NeBot l]
    (h0 : Tendsto (fun k => m ((⋃ n, s n) \ s k)) l (𝓝 0)) : m (⋃ n, s n) = ⨆ n, m (s n) := by
  set S := ⋃ n, s n
  set M := ⨆ n, m (s n)
  have hsS : ∀ {k}, s k ⊆ S := fun {k} => subset_iUnion _ _
  refine' le_antisymm _ (iSup_le fun n => m.mono hsS)
  have A : ∀ k, m S ≤ M + m (S \ s k) := fun k =>
    calc
      m S = m (s k ∪ S \ s k) := by 

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ : OuterMeasure α
ι : Type u_5
m : OuterMeasure α
s : ι → Set α
l : Filter ι
inst✝ : NeBot l
S : Set α := ⋃ n, s n
h0 : Tendsto (fun k => ↑m (S \ s k)) l (𝓝 0)
M : ℝ≥0∞ := ⨆ n, ↑m (s n)
hsS : ∀ {k : ι}, s k ⊆ S
k : ι
⊢ ↑m S = ↑m (s k ∪ S \ s k)

Next tactic:
rw [union_diff_self, union_eq_self_of_subset_left hsS]
/-- If `s : ι → Set α` is a sequence of sets, `S = ⋃ n, s n`, and `m (S \ s n)` tends to zero along
some nontrivial filter (usually `atTop` on `ι = ℕ`), then `m S = ⨆ n, m (s n)`. -/
theorem iUnion_of_tendsto_zero {ι} (m : OuterMeasure α) {s : ι → Set α} (l : Filter ι) [NeBot l]
    (h0 : Tendsto (fun k => m ((⋃ n, s n) \ s k)) l (𝓝 0)) : m (⋃ n, s n) = ⨆ n, m (s n) := by
  set S := ⋃ n, s n
  set M := ⨆ n, m (s n)
  have hsS : ∀ {k}, s k ⊆ S := fun {k} => subset_iUnion _ _
  refine' le_antisymm _ (iSup_le fun n => m.mono hsS)
  have A : ∀ k, m S ≤ M + m (S \ s k) := fun k =>
    calc
      m S = m (s k ∪ S \ s k) := by rw [union_diff_self, union_eq_self_of_subset_left hsS]
      _ ≤ m (s k) + m (S \ s k) := (m.union _ _)
      _ ≤ M + m (S \ s k) := add_le_add_right (le_iSup (m.measureOf ∘ s) k) _
  

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ : OuterMeasure α
ι : Type u_5
m : OuterMeasure α
s : ι → Set α
l : Filter ι
inst✝ : NeBot l
S : Set α := ⋃ n, s n
h0 : Tendsto (fun k => ↑m (S \ s k)) l (𝓝 0)
M : ℝ≥0∞ := ⨆ n, ↑m (s n)
hsS : ∀ {k : ι}, s k ⊆ S
A : ∀ (k : ι), ↑m S ≤ M + ↑m (S \ s k)
⊢ ↑m S ≤ M

Next tactic:
have B : Tendsto (fun k => M + m (S \ s k)) l (𝓝 (M + 0)) := tendsto_const_nhds.add h0
/-- If `s : ι → Set α` is a sequence of sets, `S = ⋃ n, s n`, and `m (S \ s n)` tends to zero along
some nontrivial filter (usually `atTop` on `ι = ℕ`), then `m S = ⨆ n, m (s n)`. -/
theorem iUnion_of_tendsto_zero {ι} (m : OuterMeasure α) {s : ι → Set α} (l : Filter ι) [NeBot l]
    (h0 : Tendsto (fun k => m ((⋃ n, s n) \ s k)) l (𝓝 0)) : m (⋃ n, s n) = ⨆ n, m (s n) := by
  set S := ⋃ n, s n
  set M := ⨆ n, m (s n)
  have hsS : ∀ {k}, s k ⊆ S := fun {k} => subset_iUnion _ _
  refine' le_antisymm _ (iSup_le fun n => m.mono hsS)
  have A : ∀ k, m S ≤ M + m (S \ s k) := fun k =>
    calc
      m S = m (s k ∪ S \ s k) := by rw [union_diff_self, union_eq_self_of_subset_left hsS]
      _ ≤ m (s k) + m (S \ s k) := (m.union _ _)
      _ ≤ M + m (S \ s k) := add_le_add_right (le_iSup (m.measureOf ∘ s) k) _
  have B : Tendsto (fun k => M + m (S \ s k)) l (𝓝 (M + 0)) := tendsto_const_nhds.add h0
  

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ : OuterMeasure α
ι : Type u_5
m : OuterMeasure α
s : ι → Set α
l : Filter ι
inst✝ : NeBot l
S : Set α := ⋃ n, s n
h0 : Tendsto (fun k => ↑m (S \ s k)) l (𝓝 0)
M : ℝ≥0∞ := ⨆ n, ↑m (s n)
hsS : ∀ {k : ι}, s k ⊆ S
A : ∀ (k : ι), ↑m S ≤ M + ↑m (S \ s k)
B : Tendsto (fun k => M + ↑m (S \ s k)) l (𝓝 (M + 0))
⊢ ↑m S ≤ M

Next tactic:
rw [add_zero] at B
/-- If `s : ι → Set α` is a sequence of sets, `S = ⋃ n, s n`, and `m (S \ s n)` tends to zero along
some nontrivial filter (usually `atTop` on `ι = ℕ`), then `m S = ⨆ n, m (s n)`. -/
theorem iUnion_of_tendsto_zero {ι} (m : OuterMeasure α) {s : ι → Set α} (l : Filter ι) [NeBot l]
    (h0 : Tendsto (fun k => m ((⋃ n, s n) \ s k)) l (𝓝 0)) : m (⋃ n, s n) = ⨆ n, m (s n) := by
  set S := ⋃ n, s n
  set M := ⨆ n, m (s n)
  have hsS : ∀ {k}, s k ⊆ S := fun {k} => subset_iUnion _ _
  refine' le_antisymm _ (iSup_le fun n => m.mono hsS)
  have A : ∀ k, m S ≤ M + m (S \ s k) := fun k =>
    calc
      m S = m (s k ∪ S \ s k) := by rw [union_diff_self, union_eq_self_of_subset_left hsS]
      _ ≤ m (s k) + m (S \ s k) := (m.union _ _)
      _ ≤ M + m (S \ s k) := add_le_add_right (le_iSup (m.measureOf ∘ s) k) _
  have B : Tendsto (fun k => M + m (S \ s k)) l (𝓝 (M + 0)) := tendsto_const_nhds.add h0
  rw [add_zero] at B
  

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ : OuterMeasure α
ι : Type u_5
m : OuterMeasure α
s : ι → Set α
l : Filter ι
inst✝ : NeBot l
S : Set α := ⋃ n, s n
h0 : Tendsto (fun k => ↑m (S \ s k)) l (𝓝 0)
M : ℝ≥0∞ := ⨆ n, ↑m (s n)
hsS : ∀ {k : ι}, s k ⊆ S
A : ∀ (k : ι), ↑m S ≤ M + ↑m (S \ s k)
B : Tendsto (fun k => M + ↑m (S \ s k)) l (𝓝 M)
⊢ ↑m S ≤ M

Next tactic:
exact ge_of_tendsto' B A
/-- If `s : ℕ → Set α` is a monotone sequence of sets such that `∑' k, m (s (k + 1) \ s k) ≠ ∞`,
then `m (⋃ n, s n) = ⨆ n, m (s n)`. -/
theorem iUnion_nat_of_monotone_of_tsum_ne_top (m : OuterMeasure α) {s : ℕ → Set α}
    (h_mono : ∀ n, s n ⊆ s (n + 1)) (h0 : (∑' k, m (s (k + 1) \ s k)) ≠ ∞)
    [∀ i:ℕ, DecidablePred (· ∈ s i)] : m (⋃ n, s n) = ⨆ n, m (s n) := by
  

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : ∑' (k : ℕ), ↑m (s (k + 1) \ s k) ≠ ⊤
inst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i
⊢ ↑m (⋃ n, s n) = ⨆ n, ↑m (s n)

Next tactic:
refine' m.iUnion_of_tendsto_zero atTop _
/-- If `s : ℕ → Set α` is a monotone sequence of sets such that `∑' k, m (s (k + 1) \ s k) ≠ ∞`,
then `m (⋃ n, s n) = ⨆ n, m (s n)`. -/
theorem iUnion_nat_of_monotone_of_tsum_ne_top (m : OuterMeasure α) {s : ℕ → Set α}
    (h_mono : ∀ n, s n ⊆ s (n + 1)) (h0 : (∑' k, m (s (k + 1) \ s k)) ≠ ∞)
    [∀ i:ℕ, DecidablePred (· ∈ s i)] : m (⋃ n, s n) = ⨆ n, m (s n) := by
  refine' m.iUnion_of_tendsto_zero atTop _
  

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : ∑' (k : ℕ), ↑m (s (k + 1) \ s k) ≠ ⊤
inst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i
⊢ Tendsto (fun k => ↑m ((⋃ n, s n) \ s k)) atTop (𝓝 0)

Next tactic:
refine' tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => _
/-- If `s : ℕ → Set α` is a monotone sequence of sets such that `∑' k, m (s (k + 1) \ s k) ≠ ∞`,
then `m (⋃ n, s n) = ⨆ n, m (s n)`. -/
theorem iUnion_nat_of_monotone_of_tsum_ne_top (m : OuterMeasure α) {s : ℕ → Set α}
    (h_mono : ∀ n, s n ⊆ s (n + 1)) (h0 : (∑' k, m (s (k + 1) \ s k)) ≠ ∞)
    [∀ i:ℕ, DecidablePred (· ∈ s i)] : m (⋃ n, s n) = ⨆ n, m (s n) := by
  refine' m.iUnion_of_tendsto_zero atTop _
  refine' tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => _
  

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : ∑' (k : ℕ), ↑m (s (k + 1) \ s k) ≠ ⊤
inst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i
n : ℕ
⊢ ↑m ((⋃ n, s n) \ s n) ≤ ∑' (k : ℕ), ↑m (s (k + n + 1) \ s (k + n))

Next tactic:
refine' (m.mono _).trans (m.iUnion _)
/-- If `s : ℕ → Set α` is a monotone sequence of sets such that `∑' k, m (s (k + 1) \ s k) ≠ ∞`,
then `m (⋃ n, s n) = ⨆ n, m (s n)`. -/
theorem iUnion_nat_of_monotone_of_tsum_ne_top (m : OuterMeasure α) {s : ℕ → Set α}
    (h_mono : ∀ n, s n ⊆ s (n + 1)) (h0 : (∑' k, m (s (k + 1) \ s k)) ≠ ∞)
    [∀ i:ℕ, DecidablePred (· ∈ s i)] : m (⋃ n, s n) = ⨆ n, m (s n) := by
  refine' m.iUnion_of_tendsto_zero atTop _
  refine' tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => _
  refine' (m.mono _).trans (m.iUnion _)
  -- Current goal: `(⋃ k, s k) \ s n ⊆ ⋃ k, s (k + n + 1) \ s (k + n)`
  

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : ∑' (k : ℕ), ↑m (s (k + 1) \ s k) ≠ ⊤
inst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i
n : ℕ
⊢ (⋃ n, s n) \ s n ⊆ ⋃ i, s (i + n + 1) \ s (i + n)

Next tactic:
have h' : Monotone s := @monotone_nat_of_le_succ (Set α) _ _ h_mono
/-- If `s : ℕ → Set α` is a monotone sequence of sets such that `∑' k, m (s (k + 1) \ s k) ≠ ∞`,
then `m (⋃ n, s n) = ⨆ n, m (s n)`. -/
theorem iUnion_nat_of_monotone_of_tsum_ne_top (m : OuterMeasure α) {s : ℕ → Set α}
    (h_mono : ∀ n, s n ⊆ s (n + 1)) (h0 : (∑' k, m (s (k + 1) \ s k)) ≠ ∞)
    [∀ i:ℕ, DecidablePred (· ∈ s i)] : m (⋃ n, s n) = ⨆ n, m (s n) := by
  refine' m.iUnion_of_tendsto_zero atTop _
  refine' tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => _
  refine' (m.mono _).trans (m.iUnion _)
  -- Current goal: `(⋃ k, s k) \ s n ⊆ ⋃ k, s (k + n + 1) \ s (k + n)`
  have h' : Monotone s := @monotone_nat_of_le_succ (Set α) _ _ h_mono
  

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : ∑' (k : ℕ), ↑m (s (k + 1) \ s k) ≠ ⊤
inst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i
n : ℕ
h' : Monotone s
⊢ (⋃ n, s n) \ s n ⊆ ⋃ i, s (i + n + 1) \ s (i + n)

Next tactic:
simp only [diff_subset_iff, iUnion_subset_iff]
/-- If `s : ℕ → Set α` is a monotone sequence of sets such that `∑' k, m (s (k + 1) \ s k) ≠ ∞`,
then `m (⋃ n, s n) = ⨆ n, m (s n)`. -/
theorem iUnion_nat_of_monotone_of_tsum_ne_top (m : OuterMeasure α) {s : ℕ → Set α}
    (h_mono : ∀ n, s n ⊆ s (n + 1)) (h0 : (∑' k, m (s (k + 1) \ s k)) ≠ ∞)
    [∀ i:ℕ, DecidablePred (· ∈ s i)] : m (⋃ n, s n) = ⨆ n, m (s n) := by
  refine' m.iUnion_of_tendsto_zero atTop _
  refine' tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => _
  refine' (m.mono _).trans (m.iUnion _)
  -- Current goal: `(⋃ k, s k) \ s n ⊆ ⋃ k, s (k + n + 1) \ s (k + n)`
  have h' : Monotone s := @monotone_nat_of_le_succ (Set α) _ _ h_mono
  simp only [diff_subset_iff, iUnion_subset_iff]
  

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : ∑' (k : ℕ), ↑m (s (k + 1) \ s k) ≠ ⊤
inst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i
n : ℕ
h' : Monotone s
⊢ ∀ (i : ℕ), s i ⊆ s n ∪ ⋃ i, s (i + n + 1) \ s (i + n)

Next tactic:
intro i x hx
/-- If `s : ℕ → Set α` is a monotone sequence of sets such that `∑' k, m (s (k + 1) \ s k) ≠ ∞`,
then `m (⋃ n, s n) = ⨆ n, m (s n)`. -/
theorem iUnion_nat_of_monotone_of_tsum_ne_top (m : OuterMeasure α) {s : ℕ → Set α}
    (h_mono : ∀ n, s n ⊆ s (n + 1)) (h0 : (∑' k, m (s (k + 1) \ s k)) ≠ ∞)
    [∀ i:ℕ, DecidablePred (· ∈ s i)] : m (⋃ n, s n) = ⨆ n, m (s n) := by
  refine' m.iUnion_of_tendsto_zero atTop _
  refine' tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => _
  refine' (m.mono _).trans (m.iUnion _)
  -- Current goal: `(⋃ k, s k) \ s n ⊆ ⋃ k, s (k + n + 1) \ s (k + n)`
  have h' : Monotone s := @monotone_nat_of_le_succ (Set α) _ _ h_mono
  simp only [diff_subset_iff, iUnion_subset_iff]
  intro i x hx
  

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : ∑' (k : ℕ), ↑m (s (k + 1) \ s k) ≠ ⊤
inst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i
n : ℕ
h' : Monotone s
i : ℕ
x : α
hx : x ∈ s i
⊢ x ∈ s n ∪ ⋃ i, s (i + n + 1) \ s (i + n)

Next tactic:
have : ∃i, x ∈ s i := by exists i
/-- If `s : ℕ → Set α` is a monotone sequence of sets such that `∑' k, m (s (k + 1) \ s k) ≠ ∞`,
then `m (⋃ n, s n) = ⨆ n, m (s n)`. -/
theorem iUnion_nat_of_monotone_of_tsum_ne_top (m : OuterMeasure α) {s : ℕ → Set α}
    (h_mono : ∀ n, s n ⊆ s (n + 1)) (h0 : (∑' k, m (s (k + 1) \ s k)) ≠ ∞)
    [∀ i:ℕ, DecidablePred (· ∈ s i)] : m (⋃ n, s n) = ⨆ n, m (s n) := by
  refine' m.iUnion_of_tendsto_zero atTop _
  refine' tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => _
  refine' (m.mono _).trans (m.iUnion _)
  -- Current goal: `(⋃ k, s k) \ s n ⊆ ⋃ k, s (k + n + 1) \ s (k + n)`
  have h' : Monotone s := @monotone_nat_of_le_succ (Set α) _ _ h_mono
  simp only [diff_subset_iff, iUnion_subset_iff]
  intro i x hx
  have : ∃i, x ∈ s i := by 

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : ∑' (k : ℕ), ↑m (s (k + 1) \ s k) ≠ ⊤
inst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i
n : ℕ
h' : Monotone s
i : ℕ
x : α
hx : x ∈ s i
⊢ ∃ i, x ∈ s i

Next tactic:
exists i
/-- If `s : ℕ → Set α` is a monotone sequence of sets such that `∑' k, m (s (k + 1) \ s k) ≠ ∞`,
then `m (⋃ n, s n) = ⨆ n, m (s n)`. -/
theorem iUnion_nat_of_monotone_of_tsum_ne_top (m : OuterMeasure α) {s : ℕ → Set α}
    (h_mono : ∀ n, s n ⊆ s (n + 1)) (h0 : (∑' k, m (s (k + 1) \ s k)) ≠ ∞)
    [∀ i:ℕ, DecidablePred (· ∈ s i)] : m (⋃ n, s n) = ⨆ n, m (s n) := by
  refine' m.iUnion_of_tendsto_zero atTop _
  refine' tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => _
  refine' (m.mono _).trans (m.iUnion _)
  -- Current goal: `(⋃ k, s k) \ s n ⊆ ⋃ k, s (k + n + 1) \ s (k + n)`
  have h' : Monotone s := @monotone_nat_of_le_succ (Set α) _ _ h_mono
  simp only [diff_subset_iff, iUnion_subset_iff]
  intro i x hx
  have : ∃i, x ∈ s i := by exists i
  

Proof state:
α : Type u_1
β : Type u_2
R : Type u_3
R' : Type u_4
ms : Set (OuterMeasure α)
m✝ m : OuterMeasure α
s : ℕ → Set α
h_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)
h0 : ∑' (k : ℕ), ↑m (s (k + 1) \ s k) ≠ ⊤
inst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i
n : ℕ
h' : Monotone s
i : ℕ
x : α
hx : x ∈ s i
this : ∃ i, x ∈ s i
⊢ x ∈ s n ∪ ⋃ i, s (i + n + 1) \ s (i + n)

Next tactic:
rcases Nat.findX this with ⟨j, hj, hlt⟩