Theorem: MeasureTheory.lintegral_iInf'
File: Mathlib/MeasureTheory/Integral/Lebesgue.lean

tactic: simp_rw [← iInf_apply]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
⊢ ∫⁻ (a : α), ⨅ n, f n a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
⊢ ∫⁻ (a : α), (⨅ i, f i) a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ

tactic: let p : α → (ℕ → ℝ≥0∞) → Prop := fun _ f' => Antitone f'
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
⊢ ∫⁻ (a : α), (⨅ i, f i) a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
⊢ ∫⁻ (a : α), (⨅ i, f i) a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ

tactic: have hp : ∀ᵐ x ∂μ, p x fun i => f i x := h_anti
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
⊢ ∫⁻ (a : α), (⨅ i, f i) a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
⊢ ∫⁻ (a : α), (⨅ i, f i) a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ

tactic: rw [lintegral_congr_ae (aeSeq.iInf h_meas hp).symm]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_mono : Antitone (aeSeq h_meas p)
⊢ ∫⁻ (a : α), (⨅ i, f i) a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_mono : Antitone (aeSeq h_meas p)
⊢ ∫⁻ (a : α), (⨅ n, aeSeq h_meas p n) a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ

tactic: simp_rw [iInf_apply]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_mono : Antitone (aeSeq h_meas p)
⊢ ∫⁻ (a : α), (⨅ n, aeSeq h_meas p n) a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_mono : Antitone (aeSeq h_meas p)
⊢ ∫⁻ (a : α), ⨅ i, aeSeq h_meas p i a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ

tactic: rw [lintegral_iInf (aeSeq.measurable h_meas p) h_ae_seq_mono]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_mono : Antitone (aeSeq h_meas p)
⊢ ∫⁻ (a : α), ⨅ i, aeSeq h_meas p i a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_mono : Antitone (aeSeq h_meas p)
⊢ ⨅ n, ∫⁻ (a : α), aeSeq h_meas p n a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ

α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_mono : Antitone (aeSeq h_meas p)
⊢ ∫⁻ (a : α), aeSeq h_meas p 0 a ∂μ ≠ ⊤

tactic: intro n m hnm x
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
⊢ Antitone (aeSeq h_meas p)
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m✝ : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
n m : ℕ
hnm : n ≤ m
x : α
⊢ aeSeq h_meas p m x ≤ aeSeq h_meas p n x

tactic: by_cases hx : x ∈ aeSeqSet h_meas p
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m✝ : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
n m : ℕ
hnm : n ≤ m
x : α
⊢ aeSeq h_meas p m x ≤ aeSeq h_meas p n x
state_after: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m✝ : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
n m : ℕ
hnm : n ≤ m
x : α
hx : x ∈ aeSeqSet h_meas p
⊢ aeSeq h_meas p m x ≤ aeSeq h_meas p n x

case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m✝ : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
n m : ℕ
hnm : n ≤ m
x : α
hx : x ∉ aeSeqSet h_meas p
⊢ aeSeq h_meas p m x ≤ aeSeq h_meas p n x

tactic: exact aeSeq.prop_of_mem_aeSeqSet h_meas hx hnm
state_before: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m✝ : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
n m : ℕ
hnm : n ≤ m
x : α
hx : x ∈ aeSeqSet h_meas p
⊢ aeSeq h_meas p m x ≤ aeSeq h_meas p n x
state_after: no goals

tactic: simp only [aeSeq, hx, if_false]
state_before: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m✝ : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
n m : ℕ
hnm : n ≤ m
x : α
hx : x ∉ aeSeqSet h_meas p
⊢ aeSeq h_meas p m x ≤ aeSeq h_meas p n x
state_after: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m✝ : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
n m : ℕ
hnm : n ≤ m
x : α
hx : x ∉ aeSeqSet h_meas p
⊢ ⋯.some ≤ ⋯.some

tactic: exact le_rfl
state_before: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m✝ : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
n m : ℕ
hnm : n ≤ m
x : α
hx : x ∉ aeSeqSet h_meas p
⊢ ⋯.some ≤ ⋯.some
state_after: no goals

tactic: congr
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_mono : Antitone (aeSeq h_meas p)
⊢ ⨅ n, ∫⁻ (a : α), aeSeq h_meas p n a ∂μ = ⨅ n, ∫⁻ (a : α), f n a ∂μ
state_after: case e_s
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_mono : Antitone (aeSeq h_meas p)
⊢ (fun n => ∫⁻ (a : α), aeSeq h_meas p n a ∂μ) = fun n => ∫⁻ (a : α), f n a ∂μ

tactic: exact funext fun n ↦ lintegral_congr_ae (aeSeq.aeSeq_n_eq_fun_n_ae h_meas hp n)
state_before: case e_s
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_mono : Antitone (aeSeq h_meas p)
⊢ (fun n => ∫⁻ (a : α), aeSeq h_meas p n a ∂μ) = fun n => ∫⁻ (a : α), f n a ∂μ
state_after: no goals

tactic: rwa [lintegral_congr_ae (aeSeq.aeSeq_n_eq_fun_n_ae h_meas hp 0)]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' => Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x
h_ae_seq_mono : Antitone (aeSeq h_meas p)
⊢ ∫⁻ (a : α), aeSeq h_meas p 0 a ∂μ ≠ ⊤
state_after: no goals