l3lab/ntp-mathlib huggingface dataset:
theorem univ_le_of_forall_fin_meas_le {μ : Measure α} (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
    (C : ℝ≥0∞) {f : Set α → ℝ≥0∞} (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → f s ≤ C)
    (h_F_lim :
      ∀ S : ℕ → Set α, (∀ n, MeasurableSet[m] (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n)) :
    f univ ≤ C := by
  let S := @spanningSets _ m (μ.trim hm) _
  have hS_mono : Monotone S := @monotone_spanningSets _ m (μ.trim hm) _
  

Proof state:
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 (Measure.trim μ hm)
C : ℝ≥0∞
f : Set α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → f s ≤ C
h_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n)
S : ℕ → Set α := spanningSets (Measure.trim μ hm)
hS_mono : Monotone S
⊢ f univ ≤ C

Next tactic:
have hS_meas : ∀ n, MeasurableSet[m] (S n) := @measurable_spanningSets _ m (μ.trim hm) _
theorem univ_le_of_forall_fin_meas_le {μ : Measure α} (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
    (C : ℝ≥0∞) {f : Set α → ℝ≥0∞} (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → f s ≤ C)
    (h_F_lim :
      ∀ S : ℕ → Set α, (∀ n, MeasurableSet[m] (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n)) :
    f univ ≤ C := by
  let S := @spanningSets _ m (μ.trim hm) _
  have hS_mono : Monotone S := @monotone_spanningSets _ m (μ.trim hm) _
  have hS_meas : ∀ n, MeasurableSet[m] (S n) := @measurable_spanningSets _ m (μ.trim hm) _
  

Proof state:
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 (Measure.trim μ hm)
C : ℝ≥0∞
f : Set α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → f s ≤ C
h_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n)
S : ℕ → Set α := spanningSets (Measure.trim μ hm)
hS_mono : Monotone S
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
⊢ f univ ≤ C

Next tactic:
rw [← @iUnion_spanningSets _ m (μ.trim hm)]
theorem univ_le_of_forall_fin_meas_le {μ : Measure α} (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
    (C : ℝ≥0∞) {f : Set α → ℝ≥0∞} (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → f s ≤ C)
    (h_F_lim :
      ∀ S : ℕ → Set α, (∀ n, MeasurableSet[m] (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n)) :
    f univ ≤ C := by
  let S := @spanningSets _ m (μ.trim hm) _
  have hS_mono : Monotone S := @monotone_spanningSets _ m (μ.trim hm) _
  have hS_meas : ∀ n, MeasurableSet[m] (S n) := @measurable_spanningSets _ m (μ.trim hm) _
  rw [← @iUnion_spanningSets _ m (μ.trim hm)]
  

Proof state:
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 (Measure.trim μ hm)
C : ℝ≥0∞
f : Set α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → f s ≤ C
h_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n)
S : ℕ → Set α := spanningSets (Measure.trim μ hm)
hS_mono : Monotone S
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
⊢ f (⋃ i, spanningSets (Measure.trim μ hm) i) ≤ C

Next tactic:
refine' (h_F_lim S hS_meas hS_mono).trans _
theorem univ_le_of_forall_fin_meas_le {μ : Measure α} (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
    (C : ℝ≥0∞) {f : Set α → ℝ≥0∞} (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → f s ≤ C)
    (h_F_lim :
      ∀ S : ℕ → Set α, (∀ n, MeasurableSet[m] (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n)) :
    f univ ≤ C := by
  let S := @spanningSets _ m (μ.trim hm) _
  have hS_mono : Monotone S := @monotone_spanningSets _ m (μ.trim hm) _
  have hS_meas : ∀ n, MeasurableSet[m] (S n) := @measurable_spanningSets _ m (μ.trim hm) _
  rw [← @iUnion_spanningSets _ m (μ.trim hm)]
  refine' (h_F_lim S hS_meas hS_mono).trans _
  

Proof state:
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 (Measure.trim μ hm)
C : ℝ≥0∞
f : Set α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → f s ≤ C
h_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n)
S : ℕ → Set α := spanningSets (Measure.trim μ hm)
hS_mono : Monotone S
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
⊢ ⨆ n, f (S n) ≤ C

Next tactic:
refine' iSup_le fun n => hf (S n) (hS_meas n) _
theorem univ_le_of_forall_fin_meas_le {μ : Measure α} (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
    (C : ℝ≥0∞) {f : Set α → ℝ≥0∞} (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → f s ≤ C)
    (h_F_lim :
      ∀ S : ℕ → Set α, (∀ n, MeasurableSet[m] (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n)) :
    f univ ≤ C := by
  let S := @spanningSets _ m (μ.trim hm) _
  have hS_mono : Monotone S := @monotone_spanningSets _ m (μ.trim hm) _
  have hS_meas : ∀ n, MeasurableSet[m] (S n) := @measurable_spanningSets _ m (μ.trim hm) _
  rw [← @iUnion_spanningSets _ m (μ.trim hm)]
  refine' (h_F_lim S hS_meas hS_mono).trans _
  refine' iSup_le fun n => hf (S n) (hS_meas n) _
  

Proof state:
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 (Measure.trim μ hm)
C : ℝ≥0∞
f : Set α → ℝ≥0∞
hf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → f s ≤ C
h_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n)
S : ℕ → Set α := spanningSets (Measure.trim μ hm)
hS_mono : Monotone S
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
n : ℕ
⊢ ↑↑μ (S n) ≠ ⊤

Next tactic:
exact ((le_trim hm).trans_lt (@measure_spanningSets_lt_top _ m (μ.trim hm) _ n)).ne
/-- If the Lebesgue integral of a function is bounded by some constant on all sets with finite
measure in a sub-σ-algebra and the measure is σ-finite on that sub-σ-algebra, then the integral
over the whole space is bounded by that same constant. Version for a measurable function.
See `lintegral_le_of_forall_fin_meas_le'` for the more general `AEMeasurable` version. -/
theorem lintegral_le_of_forall_fin_meas_le_of_measurable {μ : Measure α} (hm : m ≤ m0)
    [SigmaFinite (μ.trim hm)] (C : ℝ≥0∞) {f : α → ℝ≥0∞} (hf_meas : Measurable f)
    (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → ∫⁻ x in s, f x ∂μ ≤ C) : ∫⁻ x, f x ∂μ ≤ C := by
  

Proof state:
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 (Measure.trim μ 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

Next tactic:
have : ∫⁻ x in univ, f x ∂μ = ∫⁻ x, f x ∂μ := by simp only [Measure.restrict_univ]
/-- If the Lebesgue integral of a function is bounded by some constant on all sets with finite
measure in a sub-σ-algebra and the measure is σ-finite on that sub-σ-algebra, then the integral
over the whole space is bounded by that same constant. Version for a measurable function.
See `lintegral_le_of_forall_fin_meas_le'` for the more general `AEMeasurable` version. -/
theorem lintegral_le_of_forall_fin_meas_le_of_measurable {μ : Measure α} (hm : m ≤ m0)
    [SigmaFinite (μ.trim hm)] (C : ℝ≥0∞) {f : α → ℝ≥0∞} (hf_meas : Measurable f)
    (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → ∫⁻ x in s, f x ∂μ ≤ C) : ∫⁻ x, f x ∂μ ≤ C := by
  have : ∫⁻ x in univ, f x ∂μ = ∫⁻ x, f x ∂μ := by 

Proof state:
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 (Measure.trim μ 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

Next tactic:
simp only [Measure.restrict_univ]
/-- If the Lebesgue integral of a function is bounded by some constant on all sets with finite
measure in a sub-σ-algebra and the measure is σ-finite on that sub-σ-algebra, then the integral
over the whole space is bounded by that same constant. Version for a measurable function.
See `lintegral_le_of_forall_fin_meas_le'` for the more general `AEMeasurable` version. -/
theorem lintegral_le_of_forall_fin_meas_le_of_measurable {μ : Measure α} (hm : m ≤ m0)
    [SigmaFinite (μ.trim hm)] (C : ℝ≥0∞) {f : α → ℝ≥0∞} (hf_meas : Measurable f)
    (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → ∫⁻ x in s, f x ∂μ ≤ C) : ∫⁻ x, f x ∂μ ≤ C := by
  have : ∫⁻ x in univ, f x ∂μ = ∫⁻ x, f x ∂μ := by simp only [Measure.restrict_univ]
  

Proof state:
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 (Measure.trim μ 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

Next tactic:
rw [← this]
/-- If the Lebesgue integral of a function is bounded by some constant on all sets with finite
measure in a sub-σ-algebra and the measure is σ-finite on that sub-σ-algebra, then the integral
over the whole space is bounded by that same constant. Version for a measurable function.
See `lintegral_le_of_forall_fin_meas_le'` for the more general `AEMeasurable` version. -/
theorem lintegral_le_of_forall_fin_meas_le_of_measurable {μ : Measure α} (hm : m ≤ m0)
    [SigmaFinite (μ.trim hm)] (C : ℝ≥0∞) {f : α → ℝ≥0∞} (hf_meas : Measurable f)
    (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → ∫⁻ x in s, f x ∂μ ≤ C) : ∫⁻ x, f x ∂μ ≤ C := by
  have : ∫⁻ x in univ, f x ∂μ = ∫⁻ x, f x ∂μ := by simp only [Measure.restrict_univ]
  rw [← this]
  

Proof state:
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 (Measure.trim μ 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

Next tactic:
refine' univ_le_of_forall_fin_meas_le hm C hf fun S hS_meas hS_mono => _
/-- If the Lebesgue integral of a function is bounded by some constant on all sets with finite
measure in a sub-σ-algebra and the measure is σ-finite on that sub-σ-algebra, then the integral
over the whole space is bounded by that same constant. Version for a measurable function.
See `lintegral_le_of_forall_fin_meas_le'` for the more general `AEMeasurable` version. -/
theorem lintegral_le_of_forall_fin_meas_le_of_measurable {μ : Measure α} (hm : m ≤ m0)
    [SigmaFinite (μ.trim hm)] (C : ℝ≥0∞) {f : α → ℝ≥0∞} (hf_meas : Measurable f)
    (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → ∫⁻ x in s, f x ∂μ ≤ C) : ∫⁻ x, f x ∂μ ≤ C := by
  have : ∫⁻ x in univ, f x ∂μ = ∫⁻ x, f x ∂μ := by simp only [Measure.restrict_univ]
  rw [← this]
  refine' univ_le_of_forall_fin_meas_le hm C hf fun S hS_meas hS_mono => _
  

Proof state:
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 (Measure.trim μ 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

Next tactic:
rw [← lintegral_indicator]
/-- If the Lebesgue integral of a function is bounded by some constant on all sets with finite
measure in a sub-σ-algebra and the measure is σ-finite on that sub-σ-algebra, then the integral
over the whole space is bounded by that same constant. Version for a measurable function.
See `lintegral_le_of_forall_fin_meas_le'` for the more general `AEMeasurable` version. -/
theorem lintegral_le_of_forall_fin_meas_le_of_measurable {μ : Measure α} (hm : m ≤ m0)
    [SigmaFinite (μ.trim hm)] (C : ℝ≥0∞) {f : α → ℝ≥0∞} (hf_meas : Measurable f)
    (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → ∫⁻ x in s, f x ∂μ ≤ C) : ∫⁻ x, f x ∂μ ≤ C := by
  have : ∫⁻ x in univ, f x ∂μ = ∫⁻ x, f x ∂μ := by simp only [Measure.restrict_univ]
  rw [← this]
  refine' univ_le_of_forall_fin_meas_le hm C hf fun S hS_meas hS_mono => _
  rw [← lintegral_indicator]
  

Proof state:
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 (Measure.trim μ 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), indicator (⋃ n, S n) (fun x => f x) 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 (Measure.trim μ 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)

Next tactic:
swap
/-- If the Lebesgue integral of a function is bounded by some constant on all sets with finite
measure in a sub-σ-algebra and the measure is σ-finite on that sub-σ-algebra, then the integral
over the whole space is bounded by that same constant. Version for a measurable function.
See `lintegral_le_of_forall_fin_meas_le'` for the more general `AEMeasurable` version. -/
theorem lintegral_le_of_forall_fin_meas_le_of_measurable {μ : Measure α} (hm : m ≤ m0)
    [SigmaFinite (μ.trim hm)] (C : ℝ≥0∞) {f : α → ℝ≥0∞} (hf_meas : Measurable f)
    (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → ∫⁻ x in s, f x ∂μ ≤ C) : ∫⁻ x, f x ∂μ ≤ C := by
  have : ∫⁻ x in univ, f x ∂μ = ∫⁻ x, f x ∂μ := by simp only [Measure.restrict_univ]
  rw [← this]
  refine' univ_le_of_forall_fin_meas_le hm C hf fun S hS_meas hS_mono => _
  rw [← lintegral_indicator]
  swap
  · 

Proof state:
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 (Measure.trim μ 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)

Next tactic:
exact hm (⋃ n, S n) (@MeasurableSet.iUnion _ _ m _ _ hS_meas)
/-- If the Lebesgue integral of a function is bounded by some constant on all sets with finite
measure in a sub-σ-algebra and the measure is σ-finite on that sub-σ-algebra, then the integral
over the whole space is bounded by that same constant. Version for a measurable function.
See `lintegral_le_of_forall_fin_meas_le'` for the more general `AEMeasurable` version. -/
theorem lintegral_le_of_forall_fin_meas_le_of_measurable {μ : Measure α} (hm : m ≤ m0)
    [SigmaFinite (μ.trim hm)] (C : ℝ≥0∞) {f : α → ℝ≥0∞} (hf_meas : Measurable f)
    (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → ∫⁻ x in s, f x ∂μ ≤ C) : ∫⁻ x, f x ∂μ ≤ C := by
  have : ∫⁻ x in univ, f x ∂μ = ∫⁻ x, f x ∂μ := by simp only [Measure.restrict_univ]
  rw [← this]
  refine' univ_le_of_forall_fin_meas_le hm C hf fun S hS_meas hS_mono => _
  rw [← lintegral_indicator]
  swap
  · exact hm (⋃ n, S n) (@MeasurableSet.iUnion _ _ m _ _ hS_meas)
  

Proof state:
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 (Measure.trim μ 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), indicator (⋃ n, S n) (fun x => f x) a ∂μ ≤ ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ

Next 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))]
/-- If the Lebesgue integral of a function is bounded by some constant on all sets with finite
measure in a sub-σ-algebra and the measure is σ-finite on that sub-σ-algebra, then the integral
over the whole space is bounded by that same constant. Version for a measurable function.
See `lintegral_le_of_forall_fin_meas_le'` for the more general `AEMeasurable` version. -/
theorem lintegral_le_of_forall_fin_meas_le_of_measurable {μ : Measure α} (hm : m ≤ m0)
    [SigmaFinite (μ.trim hm)] (C : ℝ≥0∞) {f : α → ℝ≥0∞} (hf_meas : Measurable f)
    (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → ∫⁻ x in s, f x ∂μ ≤ C) : ∫⁻ x, f x ∂μ ≤ C := by
  have : ∫⁻ x in univ, f x ∂μ = ∫⁻ x, f x ∂μ := by simp only [Measure.restrict_univ]
  rw [← this]
  refine' univ_le_of_forall_fin_meas_le hm C hf fun S hS_meas hS_mono => _
  rw [← lintegral_indicator]
  swap
  · exact hm (⋃ n, S n) (@MeasurableSet.iUnion _ _ m _ _ hS_meas)
  have h_integral_indicator : ⨆ n, ∫⁻ x in S n, f x ∂μ = ⨆ n, ∫⁻ x, (S n).indicator f x ∂μ := by
    

Proof state:
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 (Measure.trim μ 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), indicator (S n) f x ∂μ

Next tactic:
congr
/-- If the Lebesgue integral of a function is bounded by some constant on all sets with finite
measure in a sub-σ-algebra and the measure is σ-finite on that sub-σ-algebra, then the integral
over the whole space is bounded by that same constant. Version for a measurable function.
See `lintegral_le_of_forall_fin_meas_le'` for the more general `AEMeasurable` version. -/
theorem lintegral_le_of_forall_fin_meas_le_of_measurable {μ : Measure α} (hm : m ≤ m0)
    [SigmaFinite (μ.trim hm)] (C : ℝ≥0∞) {f : α → ℝ≥0∞} (hf_meas : Measurable f)
    (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → ∫⁻ x in s, f x ∂μ ≤ C) : ∫⁻ x, f x ∂μ ≤ C := by
  have : ∫⁻ x in univ, f x ∂μ = ∫⁻ x, f x ∂μ := by simp only [Measure.restrict_univ]
  rw [← this]
  refine' univ_le_of_forall_fin_meas_le hm C hf fun S hS_meas hS_mono => _
  rw [← lintegral_indicator]
  swap
  · exact hm (⋃ n, S n) (@MeasurableSet.iUnion _ _ m _ _ hS_meas)
  have h_integral_indicator : ⨆ n, ∫⁻ x in S n, f x ∂μ = ⨆ n, ∫⁻ x, (S n).indicator f x ∂μ := by
    congr
    

Proof state:
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 (Measure.trim μ 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), indicator (S n) f x ∂μ

Next tactic:
ext1 n
/-- If the Lebesgue integral of a function is bounded by some constant on all sets with finite
measure in a sub-σ-algebra and the measure is σ-finite on that sub-σ-algebra, then the integral
over the whole space is bounded by that same constant. Version for a measurable function.
See `lintegral_le_of_forall_fin_meas_le'` for the more general `AEMeasurable` version. -/
theorem lintegral_le_of_forall_fin_meas_le_of_measurable {μ : Measure α} (hm : m ≤ m0)
    [SigmaFinite (μ.trim hm)] (C : ℝ≥0∞) {f : α → ℝ≥0∞} (hf_meas : Measurable f)
    (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → ∫⁻ x in s, f x ∂μ ≤ C) : ∫⁻ x, f x ∂μ ≤ C := by
  have : ∫⁻ x in univ, f x ∂μ = ∫⁻ x, f x ∂μ := by simp only [Measure.restrict_univ]
  rw [← this]
  refine' univ_le_of_forall_fin_meas_le hm C hf fun S hS_meas hS_mono => _
  rw [← lintegral_indicator]
  swap
  · exact hm (⋃ n, S n) (@MeasurableSet.iUnion _ _ m _ _ hS_meas)
  have h_integral_indicator : ⨆ n, ∫⁻ x in S n, f x ∂μ = ⨆ n, ∫⁻ x, (S n).indicator f x ∂μ := by
    congr
    ext1 n
    

Proof state:
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 (Measure.trim μ 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), indicator (S n) f x ∂μ

Next tactic:
rw [lintegral_indicator _ (hm _ (hS_meas n))]
/-- If the Lebesgue integral of a function is bounded by some constant on all sets with finite
measure in a sub-σ-algebra and the measure is σ-finite on that sub-σ-algebra, then the integral
over the whole space is bounded by that same constant. Version for a measurable function.
See `lintegral_le_of_forall_fin_meas_le'` for the more general `AEMeasurable` version. -/
theorem lintegral_le_of_forall_fin_meas_le_of_measurable {μ : Measure α} (hm : m ≤ m0)
    [SigmaFinite (μ.trim hm)] (C : ℝ≥0∞) {f : α → ℝ≥0∞} (hf_meas : Measurable f)
    (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → ∫⁻ x in s, f x ∂μ ≤ C) : ∫⁻ x, f x ∂μ ≤ C := by
  have : ∫⁻ x in univ, f x ∂μ = ∫⁻ x, f x ∂μ := by simp only [Measure.restrict_univ]
  rw [← this]
  refine' univ_le_of_forall_fin_meas_le hm C hf fun S hS_meas hS_mono => _
  rw [← lintegral_indicator]
  swap
  · exact hm (⋃ n, S n) (@MeasurableSet.iUnion _ _ m _ _ hS_meas)
  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))]
  

Proof state:
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 (Measure.trim μ 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), indicator (S n) f x ∂μ
⊢ ∫⁻ (a : α), indicator (⋃ n, S n) (fun x => f x) a ∂μ ≤ ⨆ n, ∫⁻ (x : α) in S n, f x ∂μ

Next tactic:
rw [h_integral_indicator, ← lintegral_iSup]