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

Proof steps:
state_before: α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
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✝ : l.IsMeasurablyGenerated
hfm : StronglyMeasurableAtFilter f l μ
hμ : μ.FiniteAtFilter l
hf : IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)
⊢ IntegrableAtFilter f l μ
state_after: case intro
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
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✝ : l.IsMeasurablyGenerated
hfm : StronglyMeasurableAtFilter f l μ
hμ : μ.FiniteAtFilter l
hf : IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)
C : ℝ
hC : ∀ᶠ (s : Set α) in l.smallSets, ∀ x ∈ s, ‖f x‖ ≤ C
⊢ IntegrableAtFilter f l μ
tactic: obtain ⟨C, hC⟩ : ∃ C, ∀ᶠ s in l.smallSets, ∀ x ∈ s, ‖f x‖ ≤ C :=
    hf.imp fun C hC => eventually_smallSets.2 ⟨_, hC, fun t => id⟩
state_before: case intro
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
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✝ : l.IsMeasurablyGenerated
hfm : StronglyMeasurableAtFilter f l μ
hμ : μ.FiniteAtFilter l
hf : IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)
C : ℝ
hC : ∀ᶠ (s : Set α) in l.smallSets, ∀ x ∈ s, ‖f x‖ ≤ C
⊢ IntegrableAtFilter f l μ
state_after: case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
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✝ : l.IsMeasurablyGenerated
hfm✝ : StronglyMeasurableAtFilter f l μ
hμ✝ : μ.FiniteAtFilter l
hf : IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)
C : ℝ
hC✝ : ∀ᶠ (s : Set α) in l.smallSets, ∀ x ∈ s, ‖f x‖ ≤ C
s : Set α
hsl : s ∈ l
hsm : MeasurableSet s
hfm : AEStronglyMeasurable f (μ.restrict s)
hμ : μ s < ⊤
hC : ∀ x ∈ s, ‖f x‖ ≤ C
⊢ IntegrableAtFilter f l μ
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.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
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✝ : l.IsMeasurablyGenerated
hfm✝ : StronglyMeasurableAtFilter f l μ
hμ✝ : μ.FiniteAtFilter l
hf : IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)
C : ℝ
hC✝ : ∀ᶠ (s : Set α) in l.smallSets, ∀ x ∈ s, ‖f x‖ ≤ C
s : Set α
hsl : s ∈ l
hsm : MeasurableSet s
hfm : AEStronglyMeasurable f (μ.restrict s)
hμ : μ s < ⊤
hC : ∀ x ∈ s, ‖f x‖ ≤ C
⊢ IntegrableAtFilter f l μ
state_after: case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
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✝ : l.IsMeasurablyGenerated
hfm✝ : StronglyMeasurableAtFilter f l μ
hμ✝ : μ.FiniteAtFilter l
hf : IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)
C : ℝ
hC✝ : ∀ᶠ (s : Set α) in l.smallSets, ∀ x ∈ s, ‖f x‖ ≤ C
s : Set α
hsl : s ∈ l
hsm : MeasurableSet s
hfm : AEStronglyMeasurable f (μ.restrict s)
hμ : μ s < ⊤
hC : ∀ x ∈ s, ‖f x‖ ≤ C
⊢ ∀ᵐ (x : α) ∂μ.restrict s, ‖f x‖ ≤ C
tactic: refine ⟨s, hsl, ⟨hfm, hasFiniteIntegral_restrict_of_bounded hμ (C := C) ?_⟩⟩
state_before: case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
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✝ : l.IsMeasurablyGenerated
hfm✝ : StronglyMeasurableAtFilter f l μ
hμ✝ : μ.FiniteAtFilter l
hf : IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)
C : ℝ
hC✝ : ∀ᶠ (s : Set α) in l.smallSets, ∀ x ∈ s, ‖f x‖ ≤ C
s : Set α
hsl : s ∈ l
hsm : MeasurableSet s
hfm : AEStronglyMeasurable f (μ.restrict s)
hμ : μ s < ⊤
hC : ∀ x ∈ s, ‖f x‖ ≤ C
⊢ ∀ᵐ (x : α) ∂μ.restrict s, ‖f x‖ ≤ C
state_after: case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
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✝ : l.IsMeasurablyGenerated
hfm✝ : StronglyMeasurableAtFilter f l μ
hμ✝ : μ.FiniteAtFilter l
hf : IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)
C : ℝ
hC✝ : ∀ᶠ (s : Set α) in l.smallSets, ∀ x ∈ s, ‖f x‖ ≤ C
s : Set α
hsl : s ∈ l
hsm : MeasurableSet s
hfm : AEStronglyMeasurable f (μ.restrict s)
hμ : μ s < ⊤
hC : ∀ x ∈ s, ‖f x‖ ≤ C
⊢ ∀ᵐ (x : α) ∂μ, x ∈ s → ‖f x‖ ≤ C
tactic: rw [ae_restrict_eq hsm, eventually_inf_principal]
state_before: case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
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✝ : l.IsMeasurablyGenerated
hfm✝ : StronglyMeasurableAtFilter f l μ
hμ✝ : μ.FiniteAtFilter l
hf : IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ f)
C : ℝ
hC✝ : ∀ᶠ (s : Set α) in l.smallSets, ∀ x ∈ s, ‖f x‖ ≤ C
s : Set α
hsl : s ∈ l
hsm : MeasurableSet s
hfm : AEStronglyMeasurable f (μ.restrict s)
hμ : μ s < ⊤
hC : ∀ x ∈ s, ‖f x‖ ≤ C
⊢ ∀ᵐ (x : α) ∂μ, x ∈ s → ‖f x‖ ≤ C
state_after: no goals
tactic: exact eventually_of_forall hC