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

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m m0 : MeasurableSpace α
E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ C
⊢ ∫⁻ (x : α), f x ∂μ ≤ C
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m m0 : MeasurableSpace α
E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ C
this : ∫⁻ (x : α) in univ, f x ∂μ = ∫⁻ (x : α), f x ∂μ
⊢ ∫⁻ (x : α), f x ∂μ ≤ C
tactic: have : ∫⁻ x in univ, f x ∂μ = ∫⁻ x, f x ∂μ := by simp only [Measure.restrict_univ]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m m0 : MeasurableSpace α
E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ C
this : ∫⁻ (x : α) in univ, f x ∂μ = ∫⁻ (x : α), f x ∂μ
⊢ ∫⁻ (x : α), f x ∂μ ≤ C
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m m0 : MeasurableSpace α
E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ C
this : ∫⁻ (x : α) in univ, f x ∂μ = ∫⁻ (x : α), f x ∂μ
⊢ ∫⁻ (x : α) in univ, f x ∂μ ≤ C
tactic: rw [← this]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m m0 : MeasurableSpace α
E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ C
this : ∫⁻ (x : α) in univ, f x ∂μ = ∫⁻ (x : α), f x ∂μ
⊢ ∫⁻ (x : α) in univ, f x ∂μ ≤ C
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m m0 : MeasurableSpace α
E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ C
this : ∫⁻ (x : α) in univ, f x ∂μ = ∫⁻ (x : α), f x ∂μ
S : ℕ → Set α
x✝ : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
⊢ ∫⁻ (x : α) in ⋃ n, S n, f x ∂μ ≤ ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ
tactic: refine univ_le_of_forall_fin_meas_le hm C hf fun S _ hS_mono => ?_
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m m0 : MeasurableSpace α
E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ C
this : ∫⁻ (x : α) in univ, f x ∂μ = ∫⁻ (x : α), f x ∂μ
S : ℕ → Set α
x✝ : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
⊢ ∫⁻ (x : α) in ⋃ n, S n, f x ∂μ ≤ ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ
state_after: case hd
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m m0 : MeasurableSpace α
E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ C
this : ∫⁻ (x : α) in univ, f x ∂μ = ∫⁻ (x : α), f x ∂μ
S : ℕ → Set α
x✝ : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
⊢ Directed (fun x x_1 => x ⊆ x_1) S
tactic: rw [setLIntegral_iUnion_of_directed]
state_before: case hd
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m m0 : MeasurableSpace α
E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
C : ℝ≥0∞
f : α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ C
this : ∫⁻ (x : α) in univ, f x ∂μ = ∫⁻ (x : α), f x ∂μ
S : ℕ → Set α
x✝ : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
⊢ Directed (fun x x_1 => x ⊆ x_1) S
state_after: no goals
tactic: exact directed_of_isDirected_le hS_mono