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

tactic: cases isEmpty_or_nonempty α
state_before: 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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
⊢ Tendsto (fun x => μ (Iic x)) atTop (𝓝 (μ univ))
state_after: case inl
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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : IsEmpty α
⊢ Tendsto (fun x => μ (Iic x)) atTop (𝓝 (μ univ))

case inr
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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : Nonempty α
⊢ Tendsto (fun x => μ (Iic x)) atTop (𝓝 (μ univ))

tactic: have h_mono : Monotone fun x => μ (Iic x) := fun i j hij => by simp only; gcongr
state_before: case inr
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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : Nonempty α
⊢ Tendsto (fun x => μ (Iic x)) atTop (𝓝 (μ univ))
state_after: case inr
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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : Nonempty α
h_mono : Monotone fun x => μ (Iic x)
⊢ Tendsto (fun x => μ (Iic x)) atTop (𝓝 (μ univ))

tactic: convert tendsto_atTop_iSup h_mono
state_before: case inr
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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : Nonempty α
h_mono : Monotone fun x => μ (Iic x)
⊢ Tendsto (fun x => μ (Iic x)) atTop (𝓝 (μ univ))
state_after: case h.e'_5.h.e'_3
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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : Nonempty α
h_mono : Monotone fun x => μ (Iic x)
⊢ μ univ = ⨆ i, μ (Iic i)

tactic: obtain ⟨xs, hxs_mono, hxs_tendsto⟩ := exists_seq_monotone_tendsto_atTop_atTop α
state_before: case h.e'_5.h.e'_3
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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : Nonempty α
h_mono : Monotone fun x => μ (Iic x)
⊢ μ univ = ⨆ i, μ (Iic i)
state_after: case h.e'_5.h.e'_3.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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : Nonempty α
h_mono : Monotone fun x => μ (Iic x)
xs : ℕ → α
hxs_mono : Monotone xs
hxs_tendsto : Tendsto xs atTop atTop
⊢ μ univ = ⨆ i, μ (Iic i)

tactic: have h_univ : (univ : Set α) = ⋃ n, Iic (xs n) := by
  ext1 x
  simp only [mem_univ, mem_iUnion, mem_Iic, true_iff_iff]
  obtain ⟨n, hn⟩ := tendsto_atTop_atTop.mp hxs_tendsto x
  exact ⟨n, hn n le_rfl⟩
state_before: case h.e'_5.h.e'_3.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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : Nonempty α
h_mono : Monotone fun x => μ (Iic x)
xs : ℕ → α
hxs_mono : Monotone xs
hxs_tendsto : Tendsto xs atTop atTop
⊢ μ univ = ⨆ i, μ (Iic i)
state_after: case h.e'_5.h.e'_3.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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : Nonempty α
h_mono : Monotone fun x => μ (Iic x)
xs : ℕ → α
hxs_mono : Monotone xs
hxs_tendsto : Tendsto xs atTop atTop
h_univ : univ = ⋃ n, Iic (xs n)
⊢ μ univ = ⨆ i, μ (Iic i)

tactic: rw [h_univ, measure_iUnion_eq_iSup, iSup_eq_iSup_subseq_of_monotone h_mono hxs_tendsto]
state_before: case h.e'_5.h.e'_3.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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : Nonempty α
h_mono : Monotone fun x => μ (Iic x)
xs : ℕ → α
hxs_mono : Monotone xs
hxs_tendsto : Tendsto xs atTop atTop
h_univ : univ = ⋃ n, Iic (xs n)
⊢ μ univ = ⨆ i, μ (Iic i)
state_after: case h.e'_5.h.e'_3.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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : Nonempty α
h_mono : Monotone fun x => μ (Iic x)
xs : ℕ → α
hxs_mono : Monotone xs
hxs_tendsto : Tendsto xs atTop atTop
h_univ : univ = ⋃ n, Iic (xs n)
⊢ Directed (fun x x_1 => x ⊆ x_1) fun n => Iic (xs n)

tactic: exact Monotone.directed_le fun i j hij => Iic_subset_Iic.mpr (hxs_mono hij)
state_before: case h.e'_5.h.e'_3.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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : Nonempty α
h_mono : Monotone fun x => μ (Iic x)
xs : ℕ → α
hxs_mono : Monotone xs
hxs_tendsto : Tendsto xs atTop atTop
h_univ : univ = ⋃ n, Iic (xs n)
⊢ Directed (fun x x_1 => x ⊆ x_1) fun n => Iic (xs n)
state_after: no goals

tactic: have h1 : ∀ x : α, Iic x = ∅ := fun x => Subsingleton.elim _ _
state_before: case inl
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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : IsEmpty α
⊢ Tendsto (fun x => μ (Iic x)) atTop (𝓝 (μ univ))
state_after: case inl
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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : IsEmpty α
h1 : ∀ (x : α), Iic x = ∅
⊢ Tendsto (fun x => μ (Iic x)) atTop (𝓝 (μ univ))

tactic: have h2 : (univ : Set α) = ∅ := Subsingleton.elim _ _
state_before: case inl
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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : IsEmpty α
h1 : ∀ (x : α), Iic x = ∅
⊢ Tendsto (fun x => μ (Iic x)) atTop (𝓝 (μ univ))
state_after: case inl
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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : IsEmpty α
h1 : ∀ (x : α), Iic x = ∅
h2 : univ = ∅
⊢ Tendsto (fun x => μ (Iic x)) atTop (𝓝 (μ univ))

tactic: simp_rw [h1, h2]
state_before: case inl
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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : IsEmpty α
h1 : ∀ (x : α), Iic x = ∅
h2 : univ = ∅
⊢ Tendsto (fun x => μ (Iic x)) atTop (𝓝 (μ univ))
state_after: case inl
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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : IsEmpty α
h1 : ∀ (x : α), Iic x = ∅
h2 : univ = ∅
⊢ Tendsto (fun x => μ ∅) atTop (𝓝 (μ ∅))

tactic: exact tendsto_const_nhds
state_before: case inl
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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : IsEmpty α
h1 : ∀ (x : α), Iic x = ∅
h2 : univ = ∅
⊢ Tendsto (fun x => μ ∅) atTop (𝓝 (μ ∅))
state_after: no goals

tactic: simp only
state_before: 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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : Nonempty α
i j : α
hij : i ≤ j
⊢ (fun x => μ (Iic x)) i ≤ (fun x => μ (Iic x)) j
state_after: 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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : Nonempty α
i j : α
hij : i ≤ j
⊢ μ (Iic i) ≤ μ (Iic j)

tactic: gcongr
state_before: 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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : Nonempty α
i j : α
hij : i ≤ j
⊢ μ (Iic i) ≤ μ (Iic j)
state_after: no goals

tactic: ext1 x
state_before: 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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : Nonempty α
h_mono : Monotone fun x => μ (Iic x)
xs : ℕ → α
hxs_mono : Monotone xs
hxs_tendsto : Tendsto xs atTop atTop
⊢ univ = ⋃ n, Iic (xs n)
state_after: case h
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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : Nonempty α
h_mono : Monotone fun x => μ (Iic x)
xs : ℕ → α
hxs_mono : Monotone xs
hxs_tendsto : Tendsto xs atTop atTop
x : α
⊢ x ∈ univ ↔ x ∈ ⋃ n, Iic (xs n)

tactic: simp only [mem_univ, mem_iUnion, mem_Iic, true_iff_iff]
state_before: case h
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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : Nonempty α
h_mono : Monotone fun x => μ (Iic x)
xs : ℕ → α
hxs_mono : Monotone xs
hxs_tendsto : Tendsto xs atTop atTop
x : α
⊢ x ∈ univ ↔ x ∈ ⋃ n, Iic (xs n)
state_after: case h
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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : Nonempty α
h_mono : Monotone fun x => μ (Iic x)
xs : ℕ → α
hxs_mono : Monotone xs
hxs_tendsto : Tendsto xs atTop atTop
x : α
⊢ ∃ i, x ≤ xs i

tactic: obtain ⟨n, hn⟩ := tendsto_atTop_atTop.mp hxs_tendsto x
state_before: case h
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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : Nonempty α
h_mono : Monotone fun x => μ (Iic x)
xs : ℕ → α
hxs_mono : Monotone xs
hxs_tendsto : Tendsto xs atTop atTop
x : α
⊢ ∃ i, x ≤ xs i
state_after: case h.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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : Nonempty α
h_mono : Monotone fun x => μ (Iic x)
xs : ℕ → α
hxs_mono : Monotone xs
hxs_tendsto : Tendsto xs atTop atTop
x : α
n : ℕ
hn : ∀ (a : ℕ), n ≤ a → x ≤ xs a
⊢ ∃ i, x ≤ xs i

tactic: exact ⟨n, hn n le_rfl⟩
state_before: case h.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
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
inst✝¹ : SemilatticeSup α
inst✝ : atTop.IsCountablyGenerated
μ : Measure α
h✝ : Nonempty α
h_mono : Monotone fun x => μ (Iic x)
xs : ℕ → α
hxs_mono : Monotone xs
hxs_tendsto : Tendsto xs atTop atTop
x : α
n : ℕ
hn : ∀ (a : ℕ), n ≤ a → x ≤ xs a
⊢ ∃ i, x ≤ xs i
state_after: no goals