-- Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean
theorem AEFinStronglyMeasurable.ae_nonneg_of_forall_setIntegral_nonneg {f : α → ℝ}
    (hf : AEFinStronglyMeasurable f μ)
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
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:
haveI : SigmaFinite (μ.restrict t) := hf.sigmaFinite_restrict

Result:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
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

Proof state:
case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
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 ∩ t, f x ∂μ

Tactic:
rw [Measure.restrict_apply hs] at hμts

Result:
case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
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 (s ∩ t) < ⊤
⊢ 0 ≤ ∫ (x : α) in s ∩ t, f x ∂μ

Proof state:
case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
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:
rw [Measure.restrict_restrict hs]

Result:
case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
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 ∩ t, f x ∂μ

Proof state:
case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
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 < ⊤
⊢ Integrable f (μ.restrict (s ∩ t))

Tactic:
rw [Measure.restrict_apply hs] at hμts

Result:
case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
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 (s ∩ t) < ⊤
⊢ Integrable f (μ.restrict (s ∩ t))

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
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:
suffices 0 ≤ᵐ[μ.restrict t] f from
  ae_of_ae_restrict_of_ae_restrict_compl _ this hf.ae_eq_zero_compl.symm.le

Result:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
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

Proof state:
case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
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)

Tactic:
rw [IntegrableOn, Measure.restrict_restrict hs]

Result:
case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
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 < ⊤
⊢ Integrable f (μ.restrict (s ∩ t))

Proof state:
case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
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 (s ∩ t) < ⊤
⊢ 0 ≤ ∫ (x : α) in s ∩ t, f x ∂μ

Tactic:
exact hf_zero (s ∩ t) (hs.inter hf.measurableSet) hμts

Result:
no goals

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
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

Tactic:
let t := hf.sigmaFiniteSet

Result:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
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

Proof state:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
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:
refine
  ae_nonneg_of_forall_setIntegral_nonneg_of_sigmaFinite (fun s hs hμts => ?_) fun s hs hμts => ?_

Result:
case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
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
m0 : MeasurableSpace α
μ : Measure α
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