Theorem: MeasureTheory.AEFinStronglyMeasurable.ae_nonneg_of_forall_setIntegral_nonneg
File: Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean

Proof steps:
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f✝ f : α → ℝ
hf : AEFinStronglyMeasurable f μ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
⊢ 0 ≤ᶠ[ae μ] f
state_after: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t✝ : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f✝ f : α → ℝ
hf : AEFinStronglyMeasurable f μ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
t : Set α := hf.sigmaFiniteSet
⊢ 0 ≤ᶠ[ae μ] f
tactic: let t := hf.sigmaFiniteSet
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t✝ : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f✝ f : α → ℝ
hf : AEFinStronglyMeasurable f μ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
t : Set α := hf.sigmaFiniteSet
⊢ 0 ≤ᶠ[ae μ] f
state_after: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t✝ : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f✝ f : α → ℝ
hf : AEFinStronglyMeasurable f μ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
t : Set α := hf.sigmaFiniteSet
⊢ 0 ≤ᶠ[ae (μ.restrict t)] f
tactic: suffices 0 ≤ᵐ[μ.restrict t] f from
    ae_of_ae_restrict_of_ae_restrict_compl _ this hf.ae_eq_zero_compl.symm.le
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t✝ : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f✝ f : α → ℝ
hf : AEFinStronglyMeasurable f μ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
t : Set α := hf.sigmaFiniteSet
⊢ 0 ≤ᶠ[ae (μ.restrict t)] f
state_after: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t✝ : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f✝ f : α → ℝ
hf : AEFinStronglyMeasurable f μ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
t : Set α := hf.sigmaFiniteSet
this : SigmaFinite (μ.restrict t)
⊢ 0 ≤ᶠ[ae (μ.restrict t)] f
tactic: haveI : SigmaFinite (μ.restrict t) := hf.sigmaFinite_restrict
state_before: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s t✝ : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f✝ f : α → ℝ
hf : AEFinStronglyMeasurable f μ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
t : Set α := hf.sigmaFiniteSet
this : SigmaFinite (μ.restrict t)
⊢ 0 ≤ᶠ[ae (μ.restrict t)] f
state_after: case refine_1
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s✝ t✝ : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f✝ f : α → ℝ
hf : AEFinStronglyMeasurable f μ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
t : Set α := hf.sigmaFiniteSet
this : SigmaFinite (μ.restrict t)
s : Set α
hs : MeasurableSet s
hμts : (μ.restrict t) s < ⊤
⊢ IntegrableOn f s (μ.restrict t)

case refine_2
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
μ : Measure α
s✝ t✝ : Set α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
p : ℝ≥0∞
f✝ f : α → ℝ
hf : AEFinStronglyMeasurable f μ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
t : Set α := hf.sigmaFiniteSet
this : SigmaFinite (μ.restrict t)
s : Set α
hs : MeasurableSet s
hμts : (μ.restrict t) s < ⊤
⊢ 0 ≤ ∫ (x : α) in s, f x ∂μ.restrict t
tactic: refine
    ae_nonneg_of_forall_setIntegral_nonneg_of_sigmaFinite (fun s hs hμts => ?_) fun s hs hμts => ?_