Theorem: MeasureTheory.tendsto_measure_biInter_gt
File: Mathlib/MeasureTheory/Measure/MeasureSpace.lean

tactic: refine' tendsto_order.2 ⟨fun l hl => _, fun L hL => _⟩
state_before: α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r, r > a ∧ ↑↑μ (s r) ≠ ⊤
⊢ Tendsto (↑↑μ ∘ s) (𝓝[Ioi a] a) (𝓝 (↑↑μ (⋂ (r : ι) (_ : r > a), s r)))

tactic: 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⟩
state_before: case refine'_2
α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r, r > a ∧ ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : r > a), s r)
⊢ ∀ᶠ (b : ι) in 𝓝[Ioi a] a, (↑↑μ ∘ s) b < L

tactic: rw [B] at A
state_before: case refine'_2.intro.intro.intro
α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r, r > a ∧ ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : 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 : ℕ), s (u n))))
B : (⋂ (n : ℕ), s (u n)) = ⋂ (r : ι) (_ : r > a), s r
⊢ ∀ᶠ (b : ι) in 𝓝[Ioi a] a, (↑↑μ ∘ s) b < L

tactic: obtain ⟨n, hn⟩ : ∃ n, μ (s (u n)) < L := ((tendsto_order.1 A).2 _ hL).exists
state_before: case refine'_2.intro.intro.intro
α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r, r > a ∧ ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : 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 (⋂ (r : ι) (_ : r > a), s r)))
B : (⋂ (n : ℕ), s (u n)) = ⋂ (r : ι) (_ : r > a), s r
⊢ ∀ᶠ (b : ι) in 𝓝[Ioi a] a, (↑↑μ ∘ s) b < L

tactic: have : Ioc a (u n) ∈ 𝓝[>] a := Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, u_pos n⟩
state_before: case refine'_2.intro.intro.intro.intro
α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r, r > a ∧ ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : 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 (⋂ (r : ι) (_ : r > a), s r)))
B : (⋂ (n : ℕ), s (u n)) = ⋂ (r : ι) (_ : r > a), s r
n : ℕ
hn : ↑↑μ (s (u n)) < L
⊢ ∀ᶠ (b : ι) in 𝓝[Ioi a] a, (↑↑μ ∘ s) b < L

tactic: filter_upwards [this]with r hr using lt_of_le_of_lt (measure_mono (hm _ _ hr.1 hr.2)) hn
state_before: case refine'_2.intro.intro.intro.intro
α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r, r > a ∧ ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : 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 (⋂ (r : ι) (_ : r > a), s r)))
B : (⋂ (n : ℕ), s (u n)) = ⋂ (r : ι) (_ : r > a), s r
n : ℕ
hn : ↑↑μ (s (u n)) < L
this : Ioc a (u n) ∈ 𝓝[Ioi a] a
⊢ ∀ᶠ (b : ι) in 𝓝[Ioi a] a, (↑↑μ ∘ s) b < L

tactic: filter_upwards [self_mem_nhdsWithin (s:=Ioi a)] with r hr using hl.trans_le
    (measure_mono (biInter_subset_of_mem hr))
state_before: case refine'_1
α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r, r > a ∧ ↑↑μ (s r) ≠ ⊤
l : ℝ≥0∞
hl : l < ↑↑μ (⋂ (r : ι) (_ : r > a), s r)
⊢ ∀ᶠ (b : ι) in 𝓝[Ioi a] a, l < (↑↑μ ∘ s) b

tactic: rcases hf with ⟨r, ar, _⟩
state_before: α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r, r > a ∧ ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : r > a), s r)
⊢ ∃ u, StrictAnti u ∧ (∀ (n : ℕ), a < u n) ∧ Tendsto u atTop (𝓝 a)

tactic: rcases exists_seq_strictAnti_tendsto' ar with ⟨w, w_anti, w_mem, w_lim⟩
state_before: case intro.intro
α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : r > a), s r)
r : ι
ar : r > a
right✝ : ↑↑μ (s r) ≠ ⊤
⊢ ∃ u, StrictAnti u ∧ (∀ (n : ℕ), a < u n) ∧ Tendsto u atTop (𝓝 a)

tactic: exact ⟨w, w_anti, fun n => (w_mem n).1, w_lim⟩
state_before: case intro.intro.intro.intro.intro
α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : r > a), s r)
r : ι
ar : r > a
right✝ : ↑↑μ (s r) ≠ ⊤
w : ℕ → ι
w_anti : StrictAnti w
w_mem : ∀ (n : ℕ), w n ∈ Ioo a r
w_lim : Tendsto w atTop (𝓝 a)
⊢ ∃ u, StrictAnti u ∧ (∀ (n : ℕ), a < u n) ∧ Tendsto u atTop (𝓝 a)

tactic: refine' tendsto_measure_iInter (fun n => hs _ (u_pos n)) _ _
state_before: α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r, r > a ∧ ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : 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)
⊢ Tendsto (↑↑μ ∘ s ∘ u) atTop (𝓝 (↑↑μ (⋂ (n : ℕ), s (u n))))

tactic: intro m n hmn
state_before: case refine'_1
α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r, r > a ∧ ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : 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)
⊢ Antitone (s ∘ u)

tactic: exact hm _ _ (u_pos n) (u_anti.antitone hmn)
state_before: case refine'_1
α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m✝ : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r, r > a ∧ ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : 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)
m n : ℕ
hmn : m ≤ n
⊢ (s ∘ u) n ≤ (s ∘ u) m

tactic: rcases hf with ⟨r, rpos, hr⟩
state_before: case refine'_2
α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r, r > a ∧ ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : 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) ≠ ⊤

tactic: obtain ⟨n, hn⟩ : ∃ n : ℕ, u n < r := ((tendsto_order.1 u_lim).2 r rpos).exists
state_before: case refine'_2.intro.intro
α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : 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) ≠ ⊤

tactic: refine' ⟨n, ne_of_lt (lt_of_le_of_lt _ hr.lt_top)⟩
state_before: case refine'_2.intro.intro.intro
α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : 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) ≠ ⊤

tactic: exact measure_mono (hm _ _ (u_pos n) hn.le)
state_before: case refine'_2.intro.intro.intro
α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : 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)

tactic: apply Subset.antisymm
state_before: α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r, r > a ∧ ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : 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 : ℕ), s (u n))))
⊢ (⋂ (n : ℕ), s (u n)) = ⋂ (r : ι) (_ : r > a), s r

tactic: simp only [subset_iInter_iff, gt_iff_lt]
state_before: case h₁
α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r, r > a ∧ ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : 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 : ℕ), s (u n))))
⊢ (⋂ (n : ℕ), s (u n)) ⊆ ⋂ (r : ι) (_ : r > a), s r

tactic: intro r rpos
state_before: case h₁
α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r, r > a ∧ ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : 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 : ℕ), s (u n))))
⊢ ∀ (i : ι), a < i → (⋂ (n : ℕ), s (u n)) ⊆ s i

tactic: obtain ⟨n, hn⟩ : ∃ n, u n < r := ((tendsto_order.1 u_lim).2 _ rpos).exists
state_before: case h₁
α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r, r > a ∧ ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : 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 : ℕ), s (u n))))
r : ι
rpos : a < r
⊢ (⋂ (n : ℕ), s (u n)) ⊆ s r

tactic: exact Subset.trans (iInter_subset _ n) (hm (u n) r (u_pos n) hn.le)
state_before: case h₁.intro
α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r, r > a ∧ ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : 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 : ℕ), s (u n))))
r : ι
rpos : a < r
n : ℕ
hn : u n < r
⊢ (⋂ (n : ℕ), s (u n)) ⊆ s r

tactic: simp only [subset_iInter_iff, gt_iff_lt]
state_before: case h₂
α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r, r > a ∧ ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : 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 : ℕ), s (u n))))
⊢ (⋂ (r : ι) (_ : r > a), s r) ⊆ ⋂ (n : ℕ), s (u n)

tactic: intro n
state_before: case h₂
α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r, r > a ∧ ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : 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 : ℕ), s (u n))))
⊢ ∀ (i : ℕ), (⋂ (r : ι) (_ : a < r), s r) ⊆ s (u i)

tactic: apply biInter_subset_of_mem
state_before: case h₂
α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r, r > a ∧ ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : 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 : ℕ), s (u n))))
n : ℕ
⊢ (⋂ (r : ι) (_ : a < r), s r) ⊆ s (u n)

tactic: exact u_pos n
state_before: case h₂.xs
α : Type u_2
β : Type ?u.60700
γ : Type ?u.60703
δ : Type ?u.60706
ι✝ : Type ?u.60709
R : Type ?u.60712
R' : Type ?u.60715
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s✝ s₁ s₂ t : Set α
ι : Type u_1
inst✝⁴ : LinearOrder ι
inst✝³ : TopologicalSpace ι
inst✝² : OrderTopology ι
inst✝¹ : DenselyOrdered ι
inst✝ : TopologicalSpace.FirstCountableTopology ι
s : ι → Set α
a : ι
hs : ∀ (r : ι), r > a → MeasurableSet (s r)
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r, r > a ∧ ↑↑μ (s r) ≠ ⊤
L : ℝ≥0∞
hL : L > ↑↑μ (⋂ (r : ι) (_ : 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 : ℕ), s (u n))))
n : ℕ
⊢ u n ∈ fun x => Preorder.toLT.1 a x