case h
α : Type u_1
E : Type ?u.82616
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
f✝ : α → ℝ
inst✝ : SigmaFinite μ
f : α → ℝ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
⊢ ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ᵐ (x : α) ∂Measure.restrict μ s, OfNat.ofNat 0 x ≤ f x
α : Type u_1
E : Type ?u.82616
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
f✝ : α → ℝ
inst✝ : SigmaFinite μ
f : α → ℝ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
⊢ 0 ≤ᵐ[μ] f
apply ae_of_forall_measure_lt_top_ae_restrict
case h
α : Type u_1
E : Type ?u.82616
m m0 : MeasurableSpace α
μ : Measure α
s t✝ : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
f✝ : α → ℝ
inst✝ : SigmaFinite μ
f : α → ℝ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
t : Set α
t_meas : MeasurableSet t
t_lt_top : ↑↑μ t < ⊤
⊢ ∀ᵐ (x : α) ∂Measure.restrict μ t, OfNat.ofNat 0 x ≤ f x
case h
α : Type u_1
E : Type ?u.82616
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
f✝ : α → ℝ
inst✝ : SigmaFinite μ
f : α → ℝ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
⊢ ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → ∀ᵐ (x : α) ∂Measure.restrict μ s, OfNat.ofNat 0 x ≤ f x
intro t t_meas t_lt_top
case h
α : Type u_1
E : Type ?u.82616
m m0 : MeasurableSpace α
μ : Measure α
s t✝ : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
f✝ : α → ℝ
inst✝ : SigmaFinite μ
f : α → ℝ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
t : Set α
t_meas : MeasurableSet t
t_lt_top : ↑↑μ t < ⊤
⊢ ∀ (s : Set α), MeasurableSet s → ↑↑μ (s ∩ t) < ⊤ → 0 ≤ ∫ (x : α) in s ∩ t, f x ∂μ
case h
α : Type u_1
E : Type ?u.82616
m m0 : MeasurableSpace α
μ : Measure α
s t✝ : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
f✝ : α → ℝ
inst✝ : SigmaFinite μ
f : α → ℝ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
t : Set α
t_meas : MeasurableSet t
t_lt_top : ↑↑μ t < ⊤
⊢ ∀ᵐ (x : α) ∂Measure.restrict μ t, OfNat.ofNat 0 x ≤ f x
apply ae_nonneg_restrict_of_forall_set_integral_nonneg_inter (hf_int_finite t t_meas t_lt_top)
case h
α : Type u_1
E : Type ?u.82616
m m0 : MeasurableSpace α
μ : Measure α
s✝ t✝ : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
f✝ : α → ℝ
inst✝ : SigmaFinite μ
f : α → ℝ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
t : Set α
t_meas : MeasurableSet t
t_lt_top : ↑↑μ t < ⊤
s : Set α
s_meas : MeasurableSet s
a✝ : ↑↑μ (s ∩ t) < ⊤
⊢ 0 ≤ ∫ (x : α) in s ∩ t, f x ∂μ
case h
α : Type u_1
E : Type ?u.82616
m m0 : MeasurableSpace α
μ : Measure α
s t✝ : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
f✝ : α → ℝ
inst✝ : SigmaFinite μ
f : α → ℝ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
t : Set α
t_meas : MeasurableSet t
t_lt_top : ↑↑μ t < ⊤
⊢ ∀ (s : Set α), MeasurableSet s → ↑↑μ (s ∩ t) < ⊤ → 0 ≤ ∫ (x : α) in s ∩ t, f x ∂μ
intro s s_meas _
no goals
case h
α : Type u_1
E : Type ?u.82616
m m0 : MeasurableSpace α
μ : Measure α
s✝ t✝ : Set α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
p : ℝ≥0∞
f✝ : α → ℝ
inst✝ : SigmaFinite μ
f : α → ℝ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s
hf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
t : Set α
t_meas : MeasurableSet t
t_lt_top : ↑↑μ t < ⊤
s : Set α
s_meas : MeasurableSet s
a✝ : ↑↑μ (s ∩ t) < ⊤
⊢ 0 ≤ ∫ (x : α) in s ∩ t, f x ∂μ
exact
  hf_zero _ (s_meas.inter t_meas)
    (lt_of_le_of_lt (measure_mono (Set.inter_subset_right _ _)) t_lt_top)