Theorem: MeasureTheory.Measure.FiniteAtFilter.integrableAtFilter
File: Mathlib/MeasureTheory/Integral/IntegrableOn.lean

tactic: obtain ⟨C, hC⟩ : ∃ C, ∀ᶠ s in l.smallSets, ∀ x ∈ s, ‖f x‖ ≤ C
state_before: α : Type u_1
β : Type ?u.1661712
E : Type u_2
F : Type ?u.1661718
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
f g : α → E
s t : Set α
μ ν : Measure α
l✝ l' l : Filter α
inst✝ : IsMeasurablyGenerated l
hfm : StronglyMeasurableAtFilter f l
hμ : FiniteAtFilter μ l
hf : IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)
⊢ IntegrableAtFilter f l

tactic: exact hf.imp fun C hC => eventually_smallSets.2 ⟨_, hC, fun t => id⟩
state_before: α : Type u_1
β : Type ?u.1661712
E : Type u_2
F : Type ?u.1661718
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
f g : α → E
s t : Set α
μ ν : Measure α
l✝ l' l : Filter α
inst✝ : IsMeasurablyGenerated l
hfm : StronglyMeasurableAtFilter f l
hμ : FiniteAtFilter μ l
hf : IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)
⊢ ∃ C, ∀ᶠ (s : Set α) in smallSets l, ∀ (x : α), x ∈ s → ‖f x‖ ≤ C

case intro
α : Type u_1
β : Type ?u.1661712
E : Type u_2
F : Type ?u.1661718
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
f g : α → E
s t : Set α
μ ν : Measure α
l✝ l' l : Filter α
inst✝ : IsMeasurablyGenerated l
hfm : StronglyMeasurableAtFilter f l
hμ : FiniteAtFilter μ l
hf : IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)
C : ℝ
hC : ∀ᶠ (s : Set α) in smallSets l, ∀ (x : α), x ∈ s → ‖f x‖ ≤ C
⊢ IntegrableAtFilter f l

tactic: rcases(hfm.eventually.and (hμ.eventually.and hC)).exists_measurable_mem_of_smallSets with
  ⟨s, hsl, hsm, hfm, hμ, hC⟩
state_before: case intro
α : Type u_1
β : Type ?u.1661712
E : Type u_2
F : Type ?u.1661718
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
f g : α → E
s t : Set α
μ ν : Measure α
l✝ l' l : Filter α
inst✝ : IsMeasurablyGenerated l
hfm : StronglyMeasurableAtFilter f l
hμ : FiniteAtFilter μ l
hf : IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)
C : ℝ
hC : ∀ᶠ (s : Set α) in smallSets l, ∀ (x : α), x ∈ s → ‖f x‖ ≤ C
⊢ IntegrableAtFilter f l

tactic: refine' ⟨s, hsl, ⟨hfm, hasFiniteIntegral_restrict_of_bounded hμ _⟩⟩
state_before: case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type ?u.1661712
E : Type u_2
F : Type ?u.1661718
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
f g : α → E
s✝ t : Set α
μ ν : Measure α
l✝ l' l : Filter α
inst✝ : IsMeasurablyGenerated l
hfm✝ : StronglyMeasurableAtFilter f l
hμ✝ : FiniteAtFilter μ l
hf : IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)
C : ℝ
hC✝ : ∀ᶠ (s : Set α) in smallSets l, ∀ (x : α), x ∈ s → ‖f x‖ ≤ C
s : Set α
hsl : s ∈ l
hsm : MeasurableSet s
hfm : AEStronglyMeasurable f (restrict μ s)
hμ : ↑↑μ s < ⊤
hC : ∀ (x : α), x ∈ s → ‖f x‖ ≤ C
⊢ IntegrableAtFilter f l

tactic: exact C
state_before: case intro.intro.intro.intro.intro.intro.refine'_1
α : Type u_1
β : Type ?u.1661712
E : Type u_2
F : Type ?u.1661718
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
f g : α → E
s✝ t : Set α
μ ν : Measure α
l✝ l' l : Filter α
inst✝ : IsMeasurablyGenerated l
hfm✝ : StronglyMeasurableAtFilter f l
hμ✝ : FiniteAtFilter μ l
hf : IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)
C : ℝ
hC✝ : ∀ᶠ (s : Set α) in smallSets l, ∀ (x : α), x ∈ s → ‖f x‖ ≤ C
s : Set α
hsl : s ∈ l
hsm : MeasurableSet s
hfm : AEStronglyMeasurable f (restrict μ s)
hμ : ↑↑μ s < ⊤
hC : ∀ (x : α), x ∈ s → ‖f x‖ ≤ C
⊢ ℝ

case intro.intro.intro.intro.intro.intro.refine'_2
α : Type u_1
β : Type ?u.1661712
E : Type u_2
F : Type ?u.1661718
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
f g : α → E
s✝ t : Set α
μ ν : Measure α
l✝ l' l : Filter α
inst✝ : IsMeasurablyGenerated l
hfm✝ : StronglyMeasurableAtFilter f l
hμ✝ : FiniteAtFilter μ l
hf : IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)
C : ℝ
hC✝ : ∀ᶠ (s : Set α) in smallSets l, ∀ (x : α), x ∈ s → ‖f x‖ ≤ C
s : Set α
hsl : s ∈ l
hsm : MeasurableSet s
hfm : AEStronglyMeasurable f (restrict μ s)
hμ : ↑↑μ s < ⊤
hC : ∀ (x : α), x ∈ s → ‖f x‖ ≤ C
⊢ ∀ᵐ (x : α) ∂restrict μ s, ‖f x‖ ≤ ?intro.intro.intro.intro.intro.intro.refine'_1

tactic: rw [ae_restrict_eq hsm, eventually_inf_principal]
state_before: case intro.intro.intro.intro.intro.intro.refine'_2
α : Type u_1
β : Type ?u.1661712
E : Type u_2
F : Type ?u.1661718
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
f g : α → E
s✝ t : Set α
μ ν : Measure α
l✝ l' l : Filter α
inst✝ : IsMeasurablyGenerated l
hfm✝ : StronglyMeasurableAtFilter f l
hμ✝ : FiniteAtFilter μ l
hf : IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)
C : ℝ
hC✝ : ∀ᶠ (s : Set α) in smallSets l, ∀ (x : α), x ∈ s → ‖f x‖ ≤ C
s : Set α
hsl : s ∈ l
hsm : MeasurableSet s
hfm : AEStronglyMeasurable f (restrict μ s)
hμ : ↑↑μ s < ⊤
hC : ∀ (x : α), x ∈ s → ‖f x‖ ≤ C
⊢ ∀ᵐ (x : α) ∂restrict μ s, ‖f x‖ ≤ C

tactic: exact eventually_of_forall hC
state_before: case intro.intro.intro.intro.intro.intro.refine'_2
α : Type u_1
β : Type ?u.1661712
E : Type u_2
F : Type ?u.1661718
inst✝² : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
f g : α → E
s✝ t : Set α
μ ν : Measure α
l✝ l' l : Filter α
inst✝ : IsMeasurablyGenerated l
hfm✝ : StronglyMeasurableAtFilter f l
hμ✝ : FiniteAtFilter μ l
hf : IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)
C : ℝ
hC✝ : ∀ᶠ (s : Set α) in smallSets l, ∀ (x : α), x ∈ s → ‖f x‖ ≤ C
s : Set α
hsl : s ∈ l
hsm : MeasurableSet s
hfm : AEStronglyMeasurable f (restrict μ s)
hμ : ↑↑μ s < ⊤
hC : ∀ (x : α), x ∈ s → ‖f x‖ ≤ C
⊢ ∀ᵐ (x : α) ∂μ, x ∈ s → ‖f x‖ ≤ C