-- Mathlib/MeasureTheory/Measure/MeasureSpace.lean
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, NullMeasurableSet (s r) μ) (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)))

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure 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, NullMeasurableSet (s r) μ
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, μ (s r) ≠ ⊤
⊢ atBot.IsCountablyGenerated

Tactic:
rw [← comap_coe_Ioi_nhdsGT]

Result:
α : Type u_1
m : MeasurableSpace α
μ : Measure 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, NullMeasurableSet (s r) μ
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, μ (s r) ≠ ⊤
⊢ (comap Subtype.val (𝓝[>] a)).IsCountablyGenerated

Proof state:
case hm
α : Type u_1
m : MeasurableSpace α
μ : Measure 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, NullMeasurableSet (s r) μ
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, μ (s r) ≠ ⊤
this : atBot.IsCountablyGenerated
⊢ Monotone fun x => s ↑x

Tactic:
exact fun i j h ↦ hm i j i.2 h

Result:
no goals

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure 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, NullMeasurableSet (s r) μ
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, μ (s r) ≠ ⊤
this : atBot.IsCountablyGenerated
⊢ Tendsto ((⇑μ ∘ s) ∘ Subtype.val) atBot (𝓝 (μ (⋂ x, s ↑x)))

Tactic:
apply tendsto_measure_iInter_atBot

Result:
case hs
α : Type u_1
m : MeasurableSpace α
μ : Measure 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, NullMeasurableSet (s r) μ
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, μ (s r) ≠ ⊤
this : atBot.IsCountablyGenerated
⊢ ∀ (i : { x // x ∈ Ioi a }), NullMeasurableSet (s ↑i) μ

case hm
α : Type u_1
m : MeasurableSpace α
μ : Measure 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, NullMeasurableSet (s r) μ
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, μ (s r) ≠ ⊤
this : atBot.IsCountablyGenerated
⊢ Monotone fun x => s ↑x

case hf
α : Type u_1
m : MeasurableSpace α
μ : Measure 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, NullMeasurableSet (s r) μ
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, μ (s r) ≠ ⊤
this : atBot.IsCountablyGenerated
⊢ ∃ i, μ (s ↑i) ≠ ⊤

Proof state:
case hs
α : Type u_1
m : MeasurableSpace α
μ : Measure 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, NullMeasurableSet (s r) μ
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, μ (s r) ≠ ⊤
this : atBot.IsCountablyGenerated
⊢ ∀ (i : { x // x ∈ Ioi a }), NullMeasurableSet (s ↑i) μ

Tactic:
rwa [Subtype.forall]

Result:
no goals

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure 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, NullMeasurableSet (s r) μ
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, μ (s r) ≠ ⊤
this : atBot.IsCountablyGenerated
⊢ Tendsto (⇑μ ∘ s) (𝓝[>] a) (𝓝 (μ (⋂ r, ⋂ (_ : r > a), s r)))

Tactic:
simp_rw [← map_coe_Ioi_atBot, tendsto_map'_iff, ← mem_Ioi, biInter_eq_iInter]

Result:
α : Type u_1
m : MeasurableSpace α
μ : Measure 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, NullMeasurableSet (s r) μ
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, μ (s r) ≠ ⊤
this : atBot.IsCountablyGenerated
⊢ Tendsto ((⇑μ ∘ s) ∘ Subtype.val) atBot (𝓝 (μ (⋂ x, s ↑x)))

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure 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, NullMeasurableSet (s r) μ
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, μ (s r) ≠ ⊤
⊢ Tendsto (⇑μ ∘ s) (𝓝[>] a) (𝓝 (μ (⋂ r, ⋂ (_ : r > a), s r)))

Tactic:
have : (atBot : Filter (Ioi a)).IsCountablyGenerated := by
  rw [← comap_coe_Ioi_nhdsGT]
  infer_instance

Result:
α : Type u_1
m : MeasurableSpace α
μ : Measure 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, NullMeasurableSet (s r) μ
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, μ (s r) ≠ ⊤
this : atBot.IsCountablyGenerated
⊢ Tendsto (⇑μ ∘ s) (𝓝[>] a) (𝓝 (μ (⋂ r, ⋂ (_ : r > a), s r)))

Proof state:
α : Type u_1
m : MeasurableSpace α
μ : Measure 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, NullMeasurableSet (s r) μ
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, μ (s r) ≠ ⊤
⊢ (comap Subtype.val (𝓝[>] a)).IsCountablyGenerated

Tactic:
infer_instance

Result:
no goals

Proof state:
case hf
α : Type u_1
m : MeasurableSpace α
μ : Measure 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, NullMeasurableSet (s r) μ
hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j
hf : ∃ r > a, μ (s r) ≠ ⊤
this : atBot.IsCountablyGenerated
⊢ ∃ i, μ (s ↑i) ≠ ⊤

Tactic:
simpa only [Subtype.exists, exists_prop]

Result:
no goals