l3lab/ntp-mathlib huggingface dataset:
/-- The measure of the intersection of a decreasing sequence of measurable
sets indexed by a linear order with first countable topology is the limit of the measures. -/
theorem tendsto_measure_biInter_gt {ι : Type*} [LinearOrder ι] [TopologicalSpace ι]
    [OrderTopology ι] [DenselyOrdered ι] [FirstCountableTopology ι] {s : ι → Set α}
    {a : ι} (hs : ∀ r > a, MeasurableSet (s r)) (hm : ∀ i j, a < i → i ≤ j → s i ⊆ s j)
    (hf : ∃ r > a, μ (s r) ≠ ∞) : Tendsto (μ ∘ s) (𝓝[Ioi a] a) (𝓝 (μ (⋂ r > a, s r))) := by
  refine' tendsto_order.2 ⟨fun l hl => _, fun L hL => _⟩
  · filter_upwards [self_mem_nhdsWithin (s := Ioi a)] with r hr using hl.trans_le
        (measure_mono (biInter_subset_of_mem hr))
  obtain ⟨u, u_anti, u_pos, u_lim⟩ :
    ∃ u : ℕ → ι, StrictAnti u ∧ (∀ n : ℕ, a < u n) ∧ Tendsto u atTop (𝓝 a) := by
    rcases hf with ⟨r, ar, _⟩
    rcases exists_seq_strictAnti_tendsto' ar with ⟨w, w_anti, w_mem, w_lim⟩
    exact ⟨w, w_anti, fun n => (w_mem n).1, w_lim⟩
  have A : Tendsto (μ ∘ s ∘ u) atTop (𝓝 (μ (⋂ n, s (u n)))) := by
    refine' tendsto_measure_iInter (fun n => hs _ (u_pos n)) _ _
    · intro m n hmn
      exact hm _ _ (u_pos n) (u_anti.antitone hmn)
    · 

Proof state:
case refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_8
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ r > a, MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ r, ⋂ (_ : r > a), s r)
u : ℕ → ι
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), a < u n
u_lim : Tendsto u atTop (𝓝 a)
⊢ ∃ i, ↑↑μ ((s ∘ u) i) ≠ ⊤

Next tactic:
rcases hf with ⟨r, rpos, hr⟩
/-- The measure of the intersection of a decreasing sequence of measurable
sets indexed by a linear order with first countable topology is the limit of the measures. -/
theorem tendsto_measure_biInter_gt {ι : Type*} [LinearOrder ι] [TopologicalSpace ι]
    [OrderTopology ι] [DenselyOrdered ι] [FirstCountableTopology ι] {s : ι → Set α}
    {a : ι} (hs : ∀ r > a, MeasurableSet (s r)) (hm : ∀ i j, a < i → i ≤ j → s i ⊆ s j)
    (hf : ∃ r > a, μ (s r) ≠ ∞) : Tendsto (μ ∘ s) (𝓝[Ioi a] a) (𝓝 (μ (⋂ r > a, s r))) := by
  refine' tendsto_order.2 ⟨fun l hl => _, fun L hL => _⟩
  · filter_upwards [self_mem_nhdsWithin (s := Ioi a)] with r hr using hl.trans_le
        (measure_mono (biInter_subset_of_mem hr))
  obtain ⟨u, u_anti, u_pos, u_lim⟩ :
    ∃ u : ℕ → ι, StrictAnti u ∧ (∀ n : ℕ, a < u n) ∧ Tendsto u atTop (𝓝 a) := by
    rcases hf with ⟨r, ar, _⟩
    rcases exists_seq_strictAnti_tendsto' ar with ⟨w, w_anti, w_mem, w_lim⟩
    exact ⟨w, w_anti, fun n => (w_mem n).1, w_lim⟩
  have A : Tendsto (μ ∘ s ∘ u) atTop (𝓝 (μ (⋂ n, s (u n)))) := by
    refine' tendsto_measure_iInter (fun n => hs _ (u_pos n)) _ _
    · intro m n hmn
      exact hm _ _ (u_pos n) (u_anti.antitone hmn)
    · rcases hf with ⟨r, rpos, hr⟩
      

Proof state:
case refine'_2.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_8
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ r > a, MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ r, ⋂ (_ : r > a), s r)
u : ℕ → ι
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), a < u n
u_lim : Tendsto u atTop (𝓝 a)
r : ι
rpos : r > a
hr : ↑↑μ (s r) ≠ ⊤
⊢ ∃ i, ↑↑μ ((s ∘ u) i) ≠ ⊤

Next tactic:
obtain ⟨n, hn⟩ : ∃ n : ℕ, u n < r := ((tendsto_order.1 u_lim).2 r rpos).exists
/-- The measure of the intersection of a decreasing sequence of measurable
sets indexed by a linear order with first countable topology is the limit of the measures. -/
theorem tendsto_measure_biInter_gt {ι : Type*} [LinearOrder ι] [TopologicalSpace ι]
    [OrderTopology ι] [DenselyOrdered ι] [FirstCountableTopology ι] {s : ι → Set α}
    {a : ι} (hs : ∀ r > a, MeasurableSet (s r)) (hm : ∀ i j, a < i → i ≤ j → s i ⊆ s j)
    (hf : ∃ r > a, μ (s r) ≠ ∞) : Tendsto (μ ∘ s) (𝓝[Ioi a] a) (𝓝 (μ (⋂ r > a, s r))) := by
  refine' tendsto_order.2 ⟨fun l hl => _, fun L hL => _⟩
  · filter_upwards [self_mem_nhdsWithin (s := Ioi a)] with r hr using hl.trans_le
        (measure_mono (biInter_subset_of_mem hr))
  obtain ⟨u, u_anti, u_pos, u_lim⟩ :
    ∃ u : ℕ → ι, StrictAnti u ∧ (∀ n : ℕ, a < u n) ∧ Tendsto u atTop (𝓝 a) := by
    rcases hf with ⟨r, ar, _⟩
    rcases exists_seq_strictAnti_tendsto' ar with ⟨w, w_anti, w_mem, w_lim⟩
    exact ⟨w, w_anti, fun n => (w_mem n).1, w_lim⟩
  have A : Tendsto (μ ∘ s ∘ u) atTop (𝓝 (μ (⋂ n, s (u n)))) := by
    refine' tendsto_measure_iInter (fun n => hs _ (u_pos n)) _ _
    · intro m n hmn
      exact hm _ _ (u_pos n) (u_anti.antitone hmn)
    · rcases hf with ⟨r, rpos, hr⟩
      obtain ⟨n, hn⟩ : ∃ n : ℕ, u n < r := ((tendsto_order.1 u_lim).2 r rpos).exists
      

Proof state:
case refine'_2.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_8
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ r > a, MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ r, ⋂ (_ : r > a), s r)
u : ℕ → ι
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), a < u n
u_lim : Tendsto u atTop (𝓝 a)
r : ι
rpos : r > a
hr : ↑↑μ (s r) ≠ ⊤
n : ℕ
hn : u n < r
⊢ ∃ i, ↑↑μ ((s ∘ u) i) ≠ ⊤

Next tactic:
refine' ⟨n, ne_of_lt (lt_of_le_of_lt _ hr.lt_top)⟩
/-- The measure of the intersection of a decreasing sequence of measurable
sets indexed by a linear order with first countable topology is the limit of the measures. -/
theorem tendsto_measure_biInter_gt {ι : Type*} [LinearOrder ι] [TopologicalSpace ι]
    [OrderTopology ι] [DenselyOrdered ι] [FirstCountableTopology ι] {s : ι → Set α}
    {a : ι} (hs : ∀ r > a, MeasurableSet (s r)) (hm : ∀ i j, a < i → i ≤ j → s i ⊆ s j)
    (hf : ∃ r > a, μ (s r) ≠ ∞) : Tendsto (μ ∘ s) (𝓝[Ioi a] a) (𝓝 (μ (⋂ r > a, s r))) := by
  refine' tendsto_order.2 ⟨fun l hl => _, fun L hL => _⟩
  · filter_upwards [self_mem_nhdsWithin (s := Ioi a)] with r hr using hl.trans_le
        (measure_mono (biInter_subset_of_mem hr))
  obtain ⟨u, u_anti, u_pos, u_lim⟩ :
    ∃ u : ℕ → ι, StrictAnti u ∧ (∀ n : ℕ, a < u n) ∧ Tendsto u atTop (𝓝 a) := by
    rcases hf with ⟨r, ar, _⟩
    rcases exists_seq_strictAnti_tendsto' ar with ⟨w, w_anti, w_mem, w_lim⟩
    exact ⟨w, w_anti, fun n => (w_mem n).1, w_lim⟩
  have A : Tendsto (μ ∘ s ∘ u) atTop (𝓝 (μ (⋂ n, s (u n)))) := by
    refine' tendsto_measure_iInter (fun n => hs _ (u_pos n)) _ _
    · intro m n hmn
      exact hm _ _ (u_pos n) (u_anti.antitone hmn)
    · rcases hf with ⟨r, rpos, hr⟩
      obtain ⟨n, hn⟩ : ∃ n : ℕ, u n < r := ((tendsto_order.1 u_lim).2 r rpos).exists
      refine' ⟨n, ne_of_lt (lt_of_le_of_lt _ hr.lt_top)⟩
      

Proof state:
case refine'_2.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_8
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ r > a, MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ r, ⋂ (_ : r > a), s r)
u : ℕ → ι
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), a < u n
u_lim : Tendsto u atTop (𝓝 a)
r : ι
rpos : r > a
hr : ↑↑μ (s r) ≠ ⊤
n : ℕ
hn : u n < r
⊢ ↑↑μ ((s ∘ u) n) ≤ ↑↑μ (s r)

Next tactic:
exact measure_mono (hm _ _ (u_pos n) hn.le)
/-- The measure of the intersection of a decreasing sequence of measurable
sets indexed by a linear order with first countable topology is the limit of the measures. -/
theorem tendsto_measure_biInter_gt {ι : Type*} [LinearOrder ι] [TopologicalSpace ι]
    [OrderTopology ι] [DenselyOrdered ι] [FirstCountableTopology ι] {s : ι → Set α}
    {a : ι} (hs : ∀ r > a, MeasurableSet (s r)) (hm : ∀ i j, a < i → i ≤ j → s i ⊆ s j)
    (hf : ∃ r > a, μ (s r) ≠ ∞) : Tendsto (μ ∘ s) (𝓝[Ioi a] a) (𝓝 (μ (⋂ r > a, s r))) := by
  refine' tendsto_order.2 ⟨fun l hl => _, fun L hL => _⟩
  · filter_upwards [self_mem_nhdsWithin (s := Ioi a)] with r hr using hl.trans_le
        (measure_mono (biInter_subset_of_mem hr))
  obtain ⟨u, u_anti, u_pos, u_lim⟩ :
    ∃ u : ℕ → ι, StrictAnti u ∧ (∀ n : ℕ, a < u n) ∧ Tendsto u atTop (𝓝 a) := by
    rcases hf with ⟨r, ar, _⟩
    rcases exists_seq_strictAnti_tendsto' ar with ⟨w, w_anti, w_mem, w_lim⟩
    exact ⟨w, w_anti, fun n => (w_mem n).1, w_lim⟩
  have A : Tendsto (μ ∘ s ∘ u) atTop (𝓝 (μ (⋂ n, s (u n)))) := by
    refine' tendsto_measure_iInter (fun n => hs _ (u_pos n)) _ _
    · intro m n hmn
      exact hm _ _ (u_pos n) (u_anti.antitone hmn)
    · rcases hf with ⟨r, rpos, hr⟩
      obtain ⟨n, hn⟩ : ∃ n : ℕ, u n < r := ((tendsto_order.1 u_lim).2 r rpos).exists
      refine' ⟨n, ne_of_lt (lt_of_le_of_lt _ hr.lt_top)⟩
      exact measure_mono (hm _ _ (u_pos n) hn.le)
  

Proof state:
case refine'_2.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_8
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ r > a, MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ r, ⋂ (_ : r > a), s r)
u : ℕ → ι
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), a < u n
u_lim : Tendsto u atTop (𝓝 a)
A : Tendsto (↑↑μ ∘ s ∘ u) atTop (𝓝 (↑↑μ (⋂ n, s (u n))))
⊢ ∀ᶠ (b : ι) in 𝓝[>] a, (↑↑μ ∘ s) b < L

Next tactic:
have B : ⋂ n, s (u n) = ⋂ r > a, s r := by
    apply Subset.antisymm
    · simp only [subset_iInter_iff, gt_iff_lt]
      intro r rpos
      obtain ⟨n, hn⟩ : ∃ n, u n < r := ((tendsto_order.1 u_lim).2 _ rpos).exists
      exact Subset.trans (iInter_subset _ n) (hm (u n) r (u_pos n) hn.le)
    · simp only [subset_iInter_iff, gt_iff_lt]
      intro n
      apply biInter_subset_of_mem
      exact u_pos n
/-- The measure of the intersection of a decreasing sequence of measurable
sets indexed by a linear order with first countable topology is the limit of the measures. -/
theorem tendsto_measure_biInter_gt {ι : Type*} [LinearOrder ι] [TopologicalSpace ι]
    [OrderTopology ι] [DenselyOrdered ι] [FirstCountableTopology ι] {s : ι → Set α}
    {a : ι} (hs : ∀ r > a, MeasurableSet (s r)) (hm : ∀ i j, a < i → i ≤ j → s i ⊆ s j)
    (hf : ∃ r > a, μ (s r) ≠ ∞) : Tendsto (μ ∘ s) (𝓝[Ioi a] a) (𝓝 (μ (⋂ r > a, s r))) := by
  refine' tendsto_order.2 ⟨fun l hl => _, fun L hL => _⟩
  · filter_upwards [self_mem_nhdsWithin (s := Ioi a)] with r hr using hl.trans_le
        (measure_mono (biInter_subset_of_mem hr))
  obtain ⟨u, u_anti, u_pos, u_lim⟩ :
    ∃ u : ℕ → ι, StrictAnti u ∧ (∀ n : ℕ, a < u n) ∧ Tendsto u atTop (𝓝 a) := by
    rcases hf with ⟨r, ar, _⟩
    rcases exists_seq_strictAnti_tendsto' ar with ⟨w, w_anti, w_mem, w_lim⟩
    exact ⟨w, w_anti, fun n => (w_mem n).1, w_lim⟩
  have A : Tendsto (μ ∘ s ∘ u) atTop (𝓝 (μ (⋂ n, s (u n)))) := by
    refine' tendsto_measure_iInter (fun n => hs _ (u_pos n)) _ _
    · intro m n hmn
      exact hm _ _ (u_pos n) (u_anti.antitone hmn)
    · rcases hf with ⟨r, rpos, hr⟩
      obtain ⟨n, hn⟩ : ∃ n : ℕ, u n < r := ((tendsto_order.1 u_lim).2 r rpos).exists
      refine' ⟨n, ne_of_lt (lt_of_le_of_lt _ hr.lt_top)⟩
      exact measure_mono (hm _ _ (u_pos n) hn.le)
  have B : ⋂ n, s (u n) = ⋂ r > a, s r := by
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_8
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ r > a, MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ r, ⋂ (_ : r > a), s r)
u : ℕ → ι
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), a < u n
u_lim : Tendsto u atTop (𝓝 a)
A : Tendsto (↑↑μ ∘ s ∘ u) atTop (𝓝 (↑↑μ (⋂ n, s (u n))))
⊢ ⋂ n, s (u n) = ⋂ r, ⋂ (_ : r > a), s r

Next tactic:
apply Subset.antisymm
/-- The measure of the intersection of a decreasing sequence of measurable
sets indexed by a linear order with first countable topology is the limit of the measures. -/
theorem tendsto_measure_biInter_gt {ι : Type*} [LinearOrder ι] [TopologicalSpace ι]
    [OrderTopology ι] [DenselyOrdered ι] [FirstCountableTopology ι] {s : ι → Set α}
    {a : ι} (hs : ∀ r > a, MeasurableSet (s r)) (hm : ∀ i j, a < i → i ≤ j → s i ⊆ s j)
    (hf : ∃ r > a, μ (s r) ≠ ∞) : Tendsto (μ ∘ s) (𝓝[Ioi a] a) (𝓝 (μ (⋂ r > a, s r))) := by
  refine' tendsto_order.2 ⟨fun l hl => _, fun L hL => _⟩
  · filter_upwards [self_mem_nhdsWithin (s := Ioi a)] with r hr using hl.trans_le
        (measure_mono (biInter_subset_of_mem hr))
  obtain ⟨u, u_anti, u_pos, u_lim⟩ :
    ∃ u : ℕ → ι, StrictAnti u ∧ (∀ n : ℕ, a < u n) ∧ Tendsto u atTop (𝓝 a) := by
    rcases hf with ⟨r, ar, _⟩
    rcases exists_seq_strictAnti_tendsto' ar with ⟨w, w_anti, w_mem, w_lim⟩
    exact ⟨w, w_anti, fun n => (w_mem n).1, w_lim⟩
  have A : Tendsto (μ ∘ s ∘ u) atTop (𝓝 (μ (⋂ n, s (u n)))) := by
    refine' tendsto_measure_iInter (fun n => hs _ (u_pos n)) _ _
    · intro m n hmn
      exact hm _ _ (u_pos n) (u_anti.antitone hmn)
    · rcases hf with ⟨r, rpos, hr⟩
      obtain ⟨n, hn⟩ : ∃ n : ℕ, u n < r := ((tendsto_order.1 u_lim).2 r rpos).exists
      refine' ⟨n, ne_of_lt (lt_of_le_of_lt _ hr.lt_top)⟩
      exact measure_mono (hm _ _ (u_pos n) hn.le)
  have B : ⋂ n, s (u n) = ⋂ r > a, s r := by
    apply Subset.antisymm
    · 

Proof state:
case h₁
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_8
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ r > a, MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ r, ⋂ (_ : r > a), s r)
u : ℕ → ι
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), a < u n
u_lim : Tendsto u atTop (𝓝 a)
A : Tendsto (↑↑μ ∘ s ∘ u) atTop (𝓝 (↑↑μ (⋂ n, s (u n))))
⊢ ⋂ n, s (u n) ⊆ ⋂ r, ⋂ (_ : r > a), s r

Next tactic:
simp only [subset_iInter_iff, gt_iff_lt]
/-- The measure of the intersection of a decreasing sequence of measurable
sets indexed by a linear order with first countable topology is the limit of the measures. -/
theorem tendsto_measure_biInter_gt {ι : Type*} [LinearOrder ι] [TopologicalSpace ι]
    [OrderTopology ι] [DenselyOrdered ι] [FirstCountableTopology ι] {s : ι → Set α}
    {a : ι} (hs : ∀ r > a, MeasurableSet (s r)) (hm : ∀ i j, a < i → i ≤ j → s i ⊆ s j)
    (hf : ∃ r > a, μ (s r) ≠ ∞) : Tendsto (μ ∘ s) (𝓝[Ioi a] a) (𝓝 (μ (⋂ r > a, s r))) := by
  refine' tendsto_order.2 ⟨fun l hl => _, fun L hL => _⟩
  · filter_upwards [self_mem_nhdsWithin (s := Ioi a)] with r hr using hl.trans_le
        (measure_mono (biInter_subset_of_mem hr))
  obtain ⟨u, u_anti, u_pos, u_lim⟩ :
    ∃ u : ℕ → ι, StrictAnti u ∧ (∀ n : ℕ, a < u n) ∧ Tendsto u atTop (𝓝 a) := by
    rcases hf with ⟨r, ar, _⟩
    rcases exists_seq_strictAnti_tendsto' ar with ⟨w, w_anti, w_mem, w_lim⟩
    exact ⟨w, w_anti, fun n => (w_mem n).1, w_lim⟩
  have A : Tendsto (μ ∘ s ∘ u) atTop (𝓝 (μ (⋂ n, s (u n)))) := by
    refine' tendsto_measure_iInter (fun n => hs _ (u_pos n)) _ _
    · intro m n hmn
      exact hm _ _ (u_pos n) (u_anti.antitone hmn)
    · rcases hf with ⟨r, rpos, hr⟩
      obtain ⟨n, hn⟩ : ∃ n : ℕ, u n < r := ((tendsto_order.1 u_lim).2 r rpos).exists
      refine' ⟨n, ne_of_lt (lt_of_le_of_lt _ hr.lt_top)⟩
      exact measure_mono (hm _ _ (u_pos n) hn.le)
  have B : ⋂ n, s (u n) = ⋂ r > a, s r := by
    apply Subset.antisymm
    · simp only [subset_iInter_iff, gt_iff_lt]
      

Proof state:
case h₁
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_8
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ r > a, MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ r, ⋂ (_ : r > a), s r)
u : ℕ → ι
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), a < u n
u_lim : Tendsto u atTop (𝓝 a)
A : Tendsto (↑↑μ ∘ s ∘ u) atTop (𝓝 (↑↑μ (⋂ n, s (u n))))
⊢ ∀ (i : ι), a < i → ⋂ n, s (u n) ⊆ s i

Next tactic:
intro r rpos
/-- The measure of the intersection of a decreasing sequence of measurable
sets indexed by a linear order with first countable topology is the limit of the measures. -/
theorem tendsto_measure_biInter_gt {ι : Type*} [LinearOrder ι] [TopologicalSpace ι]
    [OrderTopology ι] [DenselyOrdered ι] [FirstCountableTopology ι] {s : ι → Set α}
    {a : ι} (hs : ∀ r > a, MeasurableSet (s r)) (hm : ∀ i j, a < i → i ≤ j → s i ⊆ s j)
    (hf : ∃ r > a, μ (s r) ≠ ∞) : Tendsto (μ ∘ s) (𝓝[Ioi a] a) (𝓝 (μ (⋂ r > a, s r))) := by
  refine' tendsto_order.2 ⟨fun l hl => _, fun L hL => _⟩
  · filter_upwards [self_mem_nhdsWithin (s := Ioi a)] with r hr using hl.trans_le
        (measure_mono (biInter_subset_of_mem hr))
  obtain ⟨u, u_anti, u_pos, u_lim⟩ :
    ∃ u : ℕ → ι, StrictAnti u ∧ (∀ n : ℕ, a < u n) ∧ Tendsto u atTop (𝓝 a) := by
    rcases hf with ⟨r, ar, _⟩
    rcases exists_seq_strictAnti_tendsto' ar with ⟨w, w_anti, w_mem, w_lim⟩
    exact ⟨w, w_anti, fun n => (w_mem n).1, w_lim⟩
  have A : Tendsto (μ ∘ s ∘ u) atTop (𝓝 (μ (⋂ n, s (u n)))) := by
    refine' tendsto_measure_iInter (fun n => hs _ (u_pos n)) _ _
    · intro m n hmn
      exact hm _ _ (u_pos n) (u_anti.antitone hmn)
    · rcases hf with ⟨r, rpos, hr⟩
      obtain ⟨n, hn⟩ : ∃ n : ℕ, u n < r := ((tendsto_order.1 u_lim).2 r rpos).exists
      refine' ⟨n, ne_of_lt (lt_of_le_of_lt _ hr.lt_top)⟩
      exact measure_mono (hm _ _ (u_pos n) hn.le)
  have B : ⋂ n, s (u n) = ⋂ r > a, s r := by
    apply Subset.antisymm
    · simp only [subset_iInter_iff, gt_iff_lt]
      intro r rpos
      

Proof state:
case h₁
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_8
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ r > a, MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ r, ⋂ (_ : r > a), s r)
u : ℕ → ι
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), a < u n
u_lim : Tendsto u atTop (𝓝 a)
A : Tendsto (↑↑μ ∘ s ∘ u) atTop (𝓝 (↑↑μ (⋂ n, s (u n))))
r : ι
rpos : a < r
⊢ ⋂ n, s (u n) ⊆ s r

Next tactic:
obtain ⟨n, hn⟩ : ∃ n, u n < r := ((tendsto_order.1 u_lim).2 _ rpos).exists
/-- The measure of the intersection of a decreasing sequence of measurable
sets indexed by a linear order with first countable topology is the limit of the measures. -/
theorem tendsto_measure_biInter_gt {ι : Type*} [LinearOrder ι] [TopologicalSpace ι]
    [OrderTopology ι] [DenselyOrdered ι] [FirstCountableTopology ι] {s : ι → Set α}
    {a : ι} (hs : ∀ r > a, MeasurableSet (s r)) (hm : ∀ i j, a < i → i ≤ j → s i ⊆ s j)
    (hf : ∃ r > a, μ (s r) ≠ ∞) : Tendsto (μ ∘ s) (𝓝[Ioi a] a) (𝓝 (μ (⋂ r > a, s r))) := by
  refine' tendsto_order.2 ⟨fun l hl => _, fun L hL => _⟩
  · filter_upwards [self_mem_nhdsWithin (s := Ioi a)] with r hr using hl.trans_le
        (measure_mono (biInter_subset_of_mem hr))
  obtain ⟨u, u_anti, u_pos, u_lim⟩ :
    ∃ u : ℕ → ι, StrictAnti u ∧ (∀ n : ℕ, a < u n) ∧ Tendsto u atTop (𝓝 a) := by
    rcases hf with ⟨r, ar, _⟩
    rcases exists_seq_strictAnti_tendsto' ar with ⟨w, w_anti, w_mem, w_lim⟩
    exact ⟨w, w_anti, fun n => (w_mem n).1, w_lim⟩
  have A : Tendsto (μ ∘ s ∘ u) atTop (𝓝 (μ (⋂ n, s (u n)))) := by
    refine' tendsto_measure_iInter (fun n => hs _ (u_pos n)) _ _
    · intro m n hmn
      exact hm _ _ (u_pos n) (u_anti.antitone hmn)
    · rcases hf with ⟨r, rpos, hr⟩
      obtain ⟨n, hn⟩ : ∃ n : ℕ, u n < r := ((tendsto_order.1 u_lim).2 r rpos).exists
      refine' ⟨n, ne_of_lt (lt_of_le_of_lt _ hr.lt_top)⟩
      exact measure_mono (hm _ _ (u_pos n) hn.le)
  have B : ⋂ n, s (u n) = ⋂ r > a, s r := by
    apply Subset.antisymm
    · simp only [subset_iInter_iff, gt_iff_lt]
      intro r rpos
      obtain ⟨n, hn⟩ : ∃ n, u n < r := ((tendsto_order.1 u_lim).2 _ rpos).exists
      

Proof state:
case h₁.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_8
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ r > a, MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ r, ⋂ (_ : r > a), s r)
u : ℕ → ι
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), a < u n
u_lim : Tendsto u atTop (𝓝 a)
A : Tendsto (↑↑μ ∘ s ∘ u) atTop (𝓝 (↑↑μ (⋂ n, s (u n))))
r : ι
rpos : a < r
n : ℕ
hn : u n < r
⊢ ⋂ n, s (u n) ⊆ s r

Next tactic:
exact Subset.trans (iInter_subset _ n) (hm (u n) r (u_pos n) hn.le)
/-- The measure of the intersection of a decreasing sequence of measurable
sets indexed by a linear order with first countable topology is the limit of the measures. -/
theorem tendsto_measure_biInter_gt {ι : Type*} [LinearOrder ι] [TopologicalSpace ι]
    [OrderTopology ι] [DenselyOrdered ι] [FirstCountableTopology ι] {s : ι → Set α}
    {a : ι} (hs : ∀ r > a, MeasurableSet (s r)) (hm : ∀ i j, a < i → i ≤ j → s i ⊆ s j)
    (hf : ∃ r > a, μ (s r) ≠ ∞) : Tendsto (μ ∘ s) (𝓝[Ioi a] a) (𝓝 (μ (⋂ r > a, s r))) := by
  refine' tendsto_order.2 ⟨fun l hl => _, fun L hL => _⟩
  · filter_upwards [self_mem_nhdsWithin (s := Ioi a)] with r hr using hl.trans_le
        (measure_mono (biInter_subset_of_mem hr))
  obtain ⟨u, u_anti, u_pos, u_lim⟩ :
    ∃ u : ℕ → ι, StrictAnti u ∧ (∀ n : ℕ, a < u n) ∧ Tendsto u atTop (𝓝 a) := by
    rcases hf with ⟨r, ar, _⟩
    rcases exists_seq_strictAnti_tendsto' ar with ⟨w, w_anti, w_mem, w_lim⟩
    exact ⟨w, w_anti, fun n => (w_mem n).1, w_lim⟩
  have A : Tendsto (μ ∘ s ∘ u) atTop (𝓝 (μ (⋂ n, s (u n)))) := by
    refine' tendsto_measure_iInter (fun n => hs _ (u_pos n)) _ _
    · intro m n hmn
      exact hm _ _ (u_pos n) (u_anti.antitone hmn)
    · rcases hf with ⟨r, rpos, hr⟩
      obtain ⟨n, hn⟩ : ∃ n : ℕ, u n < r := ((tendsto_order.1 u_lim).2 r rpos).exists
      refine' ⟨n, ne_of_lt (lt_of_le_of_lt _ hr.lt_top)⟩
      exact measure_mono (hm _ _ (u_pos n) hn.le)
  have B : ⋂ n, s (u n) = ⋂ r > a, s r := by
    apply Subset.antisymm
    · simp only [subset_iInter_iff, gt_iff_lt]
      intro r rpos
      obtain ⟨n, hn⟩ : ∃ n, u n < r := ((tendsto_order.1 u_lim).2 _ rpos).exists
      exact Subset.trans (iInter_subset _ n) (hm (u n) r (u_pos n) hn.le)
    · 

Proof state:
case h₂
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι✝ : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_8
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ r > a, MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ r, ⋂ (_ : r > a), s r)
u : ℕ → ι
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), a < u n
u_lim : Tendsto u atTop (𝓝 a)
A : Tendsto (↑↑μ ∘ s ∘ u) atTop (𝓝 (↑↑μ (⋂ n, s (u n))))
⊢ ⋂ r, ⋂ (_ : r > a), s r ⊆ ⋂ n, s (u n)

Next tactic:
simp only [subset_iInter_iff, gt_iff_lt]