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

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_meas : Measurable f
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_meas : Measurable f
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: 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_meas : Measurable f
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_meas : Measurable f
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: refine univ_le_of_forall_fin_meas_le hm C hf fun S hS_meas 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_meas : Measurable f
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_meas : Measurable f
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
hS_meas : ∀ (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: rw [← lintegral_indicator]
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_meas : Measurable f
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
hS_meas : ∀ (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: 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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
⊢ ∫⁻ (a : α), (⋃ n, S n).indicator f a ∂μ ≤ ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ

case hs
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
⊢ MeasurableSet (⋃ n, S n)

tactic: swap
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
⊢ ∫⁻ (a : α), (⋃ n, S n).indicator f a ∂μ ≤ ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ

case hs
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
⊢ MeasurableSet (⋃ n, S n)
state_after: case hs
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
⊢ MeasurableSet (⋃ n, S n)

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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
⊢ ∫⁻ (a : α), (⋃ n, S n).indicator f a ∂μ ≤ ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ

tactic: have h_integral_indicator : ⨆ n, ∫⁻ x in S n, f x ∂μ = ⨆ n, ∫⁻ x, (S n).indicator f x ∂μ := by
  congr
  ext1 n
  rw [lintegral_indicator _ (hm _ (hS_meas n))]
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
⊢ ∫⁻ (a : α), (⋃ n, S n).indicator f a ∂μ ≤ ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
⊢ ∫⁻ (a : α), (⋃ n, S n).indicator f a ∂μ ≤ ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ

tactic: rw [h_integral_indicator, ← lintegral_iSup]
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
⊢ ∫⁻ (a : α), (⋃ n, S n).indicator f a ∂μ ≤ ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
⊢ ∫⁻ (a : α), (⋃ n, S n).indicator f a ∂μ ≤ ∫⁻ (a : α), ⨆ n, (S n).indicator f a ∂μ

case hf
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
⊢ ∀ (n : ℕ), Measurable ((S n).indicator f)

case h_mono
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
⊢ Monotone fun n => (S n).indicator f

tactic: 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_meas : Measurable f
hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ C
⊢ ∫⁻ (x : α) in univ, f x ∂μ = ∫⁻ (x : α), f x ∂μ
state_after: no goals

tactic: exact hm (⋃ n, S n) (@MeasurableSet.iUnion _ _ m _ _ hS_meas)
state_before: case hs
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
⊢ MeasurableSet (⋃ n, S n)
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 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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
⊢ ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
state_after: case e_s
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
⊢ (fun n => ∫⁻ (x : α) in S n, f x ∂μ) = fun n => ∫⁻ (x : α), (S n).indicator f x ∂μ

tactic: ext1 n
state_before: case e_s
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
⊢ (fun n => ∫⁻ (x : α) in S n, f x ∂μ) = fun n => ∫⁻ (x : α), (S n).indicator f x ∂μ
state_after: case e_s.h
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
n : ℕ
⊢ ∫⁻ (x : α) in S n, f x ∂μ = ∫⁻ (x : α), (S n).indicator f x ∂μ

tactic: rw [lintegral_indicator _ (hm _ (hS_meas n))]
state_before: case e_s.h
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
n : ℕ
⊢ ∫⁻ (x : α) in S n, f x ∂μ = ∫⁻ (x : α), (S n).indicator f x ∂μ
state_after: no goals

tactic: refine le_of_eq (lintegral_congr fun x => ?_)
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
⊢ ∫⁻ (a : α), (⋃ n, S n).indicator f a ∂μ ≤ ∫⁻ (a : α), ⨆ n, (S n).indicator f a ∂μ
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
⊢ (⋃ n, S n).indicator f x = ⨆ n, (S n).indicator f x

tactic: simp_rw [indicator_apply]
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
⊢ (⋃ n, S n).indicator f x = ⨆ n, (S n).indicator f x
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
⊢ (if x ∈ ⋃ n, S n then f x else 0) = ⨆ n, if x ∈ S n then f x else 0

tactic: by_cases hx_mem : x ∈ iUnion S
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
⊢ (if x ∈ ⋃ n, S n then f x else 0) = ⨆ n, if x ∈ S n then f x else 0
state_after: case pos
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
hx_mem : x ∈ iUnion S
⊢ (if x ∈ ⋃ n, S n then f x else 0) = ⨆ n, if x ∈ S n then f x else 0

case neg
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
hx_mem : x ∉ iUnion S
⊢ (if x ∈ ⋃ n, S n then f x else 0) = ⨆ n, if x ∈ S n then f x else 0

tactic: simp only [hx_mem, if_true]
state_before: case pos
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
hx_mem : x ∈ iUnion S
⊢ (if x ∈ ⋃ n, S n then f x else 0) = ⨆ n, if x ∈ S n then f x else 0
state_after: case pos
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
hx_mem : x ∈ iUnion S
⊢ f x = ⨆ n, if x ∈ S n then f x else 0

tactic: obtain ⟨n, hxn⟩ := mem_iUnion.mp hx_mem
state_before: case pos
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
hx_mem : x ∈ iUnion S
⊢ f x = ⨆ n, if x ∈ S n then f x else 0
state_after: case pos.intro
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
hx_mem : x ∈ iUnion S
n : ℕ
hxn : x ∈ S n
⊢ f x = ⨆ n, if x ∈ S n then f x else 0

tactic: refine le_antisymm (_root_.trans ?_ (le_iSup _ n)) (iSup_le fun i => ?_)
state_before: case pos.intro
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
hx_mem : x ∈ iUnion S
n : ℕ
hxn : x ∈ S n
⊢ f x = ⨆ n, if x ∈ S n then f x else 0
state_after: case pos.intro.refine_1
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
hx_mem : x ∈ iUnion S
n : ℕ
hxn : x ∈ S n
⊢ f x ≤ if x ∈ S n then f x else 0

case pos.intro.refine_2
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
hx_mem : x ∈ iUnion S
n : ℕ
hxn : x ∈ S n
i : ℕ
⊢ (if x ∈ S i then f x else 0) ≤ f x

tactic: simp only [hxn, le_refl, if_true]
state_before: case pos.intro.refine_1
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
hx_mem : x ∈ iUnion S
n : ℕ
hxn : x ∈ S n
⊢ f x ≤ if x ∈ S n then f x else 0
state_after: no goals

tactic: by_cases hxi : x ∈ S i <;> simp [hxi]
state_before: case pos.intro.refine_2
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
hx_mem : x ∈ iUnion S
n : ℕ
hxn : x ∈ S n
i : ℕ
⊢ (if x ∈ S i then f x else 0) ≤ f x
state_after: no goals

tactic: simp only [hx_mem, if_false]
state_before: case neg
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
hx_mem : x ∉ iUnion S
⊢ (if x ∈ ⋃ n, S n then f x else 0) = ⨆ n, if x ∈ S n then f x else 0
state_after: case neg
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
hx_mem : x ∉ iUnion S
⊢ 0 = ⨆ n, if x ∈ S n then f x else 0

tactic: rw [mem_iUnion] at hx_mem
state_before: case neg
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
hx_mem : x ∉ iUnion S
⊢ 0 = ⨆ n, if x ∈ S n then f x else 0
state_after: case neg
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
hx_mem : ¬∃ i, x ∈ S i
⊢ 0 = ⨆ n, if x ∈ S n then f x else 0

tactic: push_neg at hx_mem
state_before: case neg
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
hx_mem : ¬∃ i, x ∈ S i
⊢ 0 = ⨆ n, if x ∈ S n then f x else 0
state_after: case neg
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
hx_mem : ∀ (i : ℕ), x ∉ S i
⊢ 0 = ⨆ n, if x ∈ S n then f x else 0

tactic: refine le_antisymm (zero_le _) (iSup_le fun n => ?_)
state_before: case neg
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
hx_mem : ∀ (i : ℕ), x ∉ S i
⊢ 0 = ⨆ n, if x ∈ S n then f x else 0
state_after: case neg
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
hx_mem : ∀ (i : ℕ), x ∉ S i
n : ℕ
⊢ (if x ∈ S n then f x else 0) ≤ 0

tactic: simp only [hx_mem n, if_false, nonpos_iff_eq_zero]
state_before: case neg
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
x : α
hx_mem : ∀ (i : ℕ), x ∉ S i
n : ℕ
⊢ (if x ∈ S n then f x else 0) ≤ 0
state_after: no goals

tactic: exact fun n => hf_meas.indicator (hm _ (hS_meas n))
state_before: case hf
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
⊢ ∀ (n : ℕ), Measurable ((S n).indicator f)
state_after: no goals

tactic: intro n₁ n₂ hn₁₂ a
state_before: case h_mono
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
⊢ Monotone fun n => (S n).indicator f
state_after: case h_mono
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
n₁ n₂ : ℕ
hn₁₂ : n₁ ≤ n₂
a : α
⊢ (fun n => (S n).indicator f) n₁ a ≤ (fun n => (S n).indicator f) n₂ a

tactic: simp_rw [indicator_apply]
state_before: case h_mono
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
n₁ n₂ : ℕ
hn₁₂ : n₁ ≤ n₂
a : α
⊢ (fun n => (S n).indicator f) n₁ a ≤ (fun n => (S n).indicator f) n₂ a
state_after: case h_mono
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
n₁ n₂ : ℕ
hn₁₂ : n₁ ≤ n₂
a : α
⊢ (if a ∈ S n₁ then f a else 0) ≤ if a ∈ S n₂ then f a else 0

tactic: split_ifs with h h_1
state_before: case h_mono
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
n₁ n₂ : ℕ
hn₁₂ : n₁ ≤ n₂
a : α
⊢ (if a ∈ S n₁ then f a else 0) ≤ if a ∈ S n₂ then f a else 0
state_after: case pos
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
n₁ n₂ : ℕ
hn₁₂ : n₁ ≤ n₂
a : α
h : a ∈ S n₁
h_1 : a ∈ S n₂
⊢ f a ≤ f a

case neg
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
n₁ n₂ : ℕ
hn₁₂ : n₁ ≤ n₂
a : α
h : a ∈ S n₁
h_1 : a ∉ S n₂
⊢ f a ≤ 0

case pos
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
n₁ n₂ : ℕ
hn₁₂ : n₁ ≤ n₂
a : α
h : a ∉ S n₁
h✝ : a ∈ S n₂
⊢ 0 ≤ f a

case neg
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
n₁ n₂ : ℕ
hn₁₂ : n₁ ≤ n₂
a : α
h : a ∉ S n₁
h✝ : a ∉ S n₂
⊢ 0 ≤ 0

tactic: exact le_rfl
state_before: case pos
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
n₁ n₂ : ℕ
hn₁₂ : n₁ ≤ n₂
a : α
h : a ∈ S n₁
h_1 : a ∈ S n₂
⊢ f a ≤ f a
state_after: no goals

tactic: exact absurd (mem_of_mem_of_subset h (hS_mono hn₁₂)) h_1
state_before: case neg
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
n₁ n₂ : ℕ
hn₁₂ : n₁ ≤ n₂
a : α
h : a ∈ S n₁
h_1 : a ∉ S n₂
⊢ f a ≤ 0
state_after: no goals

tactic: exact zero_le _
state_before: case pos
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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
n₁ n₂ : ℕ
hn₁₂ : n₁ ≤ n₂
a : α
h : a ∉ S n₁
h✝ : a ∈ S n₂
⊢ 0 ≤ f a
state_after: no goals

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 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_meas : Measurable f
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
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
hS_mono : Monotone S
h_integral_indicator : ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ = ⨆ n, ∫⁻ (x : α), (S n).indicator f x ∂μ
n₁ n₂ : ℕ
hn₁₂ : n₁ ≤ n₂
a : α
h : a ∉ S n₁
h✝ : a ∉ S n₂
⊢ 0 ≤ 0
state_after: no goals