Theorem: MeasureTheory.limsup_lintegral_le
File: Mathlib/MeasureTheory/Integral/Lebesgue.lean

tactic: refine (lintegral_iInf ?_ ?_ ?_).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∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᶠ[ae μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤
⊢ ⨅ n, ∫⁻ (a : α), ⨆ i, ⨆ (_ : i ≥ n), f i a ∂μ = ∫⁻ (a : α), ⨅ n, ⨆ i, ⨆ (_ : i ≥ n), f i a ∂μ
state_after: case refine_1
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∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᶠ[ae μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤
⊢ ∀ (n : ℕ), Measurable fun a => ⨆ i, ⨆ (_ : i ≥ n), f i a

case refine_2
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∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᶠ[ae μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤
⊢ Antitone fun n a => ⨆ i, ⨆ (_ : i ≥ n), f i a

case refine_3
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∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᶠ[ae μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤
⊢ ∫⁻ (a : α), ⨆ i, ⨆ (_ : i ≥ 0), f i a ∂μ ≠ ⊤

tactic: intro n
state_before: case refine_1
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∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᶠ[ae μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤
⊢ ∀ (n : ℕ), Measurable fun a => ⨆ i, ⨆ (_ : i ≥ n), f i a
state_after: case refine_1
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∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᶠ[ae μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤
n : ℕ
⊢ Measurable fun a => ⨆ i, ⨆ (_ : i ≥ n), f i a

tactic: exact measurable_biSup _ (to_countable _) (fun i _ ↦ hf_meas i)
state_before: case refine_1
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∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᶠ[ae μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤
n : ℕ
⊢ Measurable fun a => ⨆ i, ⨆ (_ : i ≥ n), f i a
state_after: no goals

tactic: intro n m hnm a
state_before: case refine_2
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∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᶠ[ae μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤
⊢ Antitone fun n a => ⨆ i, ⨆ (_ : i ≥ n), f i a
state_after: case refine_2
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∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᶠ[ae μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤
n m : ℕ
hnm : n ≤ m
a : α
⊢ (fun n a => ⨆ i, ⨆ (_ : i ≥ n), f i a) m a ≤ (fun n a => ⨆ i, ⨆ (_ : i ≥ n), f i a) n a

tactic: exact iSup_le_iSup_of_subset fun i hi => le_trans hnm hi
state_before: case refine_2
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∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᶠ[ae μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤
n m : ℕ
hnm : n ≤ m
a : α
⊢ (fun n a => ⨆ i, ⨆ (_ : i ≥ n), f i a) m a ≤ (fun n a => ⨆ i, ⨆ (_ : i ≥ n), f i a) n a
state_after: no goals

tactic: refine ne_top_of_le_ne_top h_fin (lintegral_mono_ae ?_)
state_before: case refine_3
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∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᶠ[ae μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤
⊢ ∫⁻ (a : α), ⨆ i, ⨆ (_ : i ≥ 0), f i a ∂μ ≠ ⊤
state_after: case refine_3
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∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᶠ[ae μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤
⊢ ∀ᵐ (a : α) ∂μ, ⨆ i, ⨆ (_ : i ≥ 0), f i a ≤ g a

tactic: refine (ae_all_iff.2 h_bound).mono fun n hn => ?_
state_before: case refine_3
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∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᶠ[ae μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤
⊢ ∀ᵐ (a : α) ∂μ, ⨆ i, ⨆ (_ : i ≥ 0), f i a ≤ g a
state_after: case refine_3
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∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᶠ[ae μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤
n : α
hn : ∀ (i : ℕ), f i n ≤ g n
⊢ ⨆ i, ⨆ (_ : i ≥ 0), f i n ≤ g n

tactic: exact iSup_le fun i => iSup_le fun _ => hn i
state_before: case refine_3
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∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᶠ[ae μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤
n : α
hn : ∀ (i : ℕ), f i n ≤ g n
⊢ ⨆ i, ⨆ (_ : i ≥ 0), f i n ≤ g n
state_after: no goals

tactic: simp only [limsup_eq_iInf_iSup_of_nat]
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∞
g : α → ℝ≥0∞
hf_meas : ∀ (n : ℕ), Measurable (f n)
h_bound : ∀ (n : ℕ), f n ≤ᶠ[ae μ] g
h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤
⊢ ∫⁻ (a : α), ⨅ n, ⨆ i, ⨆ (_ : i ≥ n), f i a ∂μ = ∫⁻ (a : α), limsup (fun n => f n a) atTop ∂μ
state_after: no goals