Theorem: MeasureTheory.LocallyIntegrable.integrable_of_isBigO_atBot_atTop
File: Mathlib/MeasureTheory/Integral/Asymptotics.lean

Proof steps:
state_before: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁸ : MeasurableSpace α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
f : α → E
g : α → F
a b : α
μ : Measure α
l : Filter α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : CompactIccSpace α
g' : α → F
inst✝¹ : atBot.IsMeasurablyGenerated
inst✝ : atTop.IsMeasurablyGenerated
hf : LocallyIntegrable f μ
ho : f =O[atBot] g
hg : IntegrableAtFilter g atBot μ
ho' : f =O[atTop] g'
hg' : IntegrableAtFilter g' atTop μ
⊢ Integrable f μ
state_after: case refine_1
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁸ : MeasurableSpace α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
f : α → E
g : α → F
a b : α
μ : Measure α
l : Filter α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : CompactIccSpace α
g' : α → F
inst✝¹ : atBot.IsMeasurablyGenerated
inst✝ : atTop.IsMeasurablyGenerated
hf : LocallyIntegrable f μ
ho : f =O[atBot] g
hg : IntegrableAtFilter g atBot μ
ho' : f =O[atTop] g'
hg' : IntegrableAtFilter g' atTop μ
⊢ StronglyMeasurableAtFilter f atBot μ

case refine_2
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁸ : MeasurableSpace α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
f : α → E
g : α → F
a b : α
μ : Measure α
l : Filter α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : CompactIccSpace α
g' : α → F
inst✝¹ : atBot.IsMeasurablyGenerated
inst✝ : atTop.IsMeasurablyGenerated
hf : LocallyIntegrable f μ
ho : f =O[atBot] g
hg : IntegrableAtFilter g atBot μ
ho' : f =O[atTop] g'
hg' : IntegrableAtFilter g' atTop μ
⊢ StronglyMeasurableAtFilter f atTop μ
tactic: refine integrable_iff_integrableAtFilter_atBot_atTop.mpr
    ⟨⟨ho.integrableAtFilter ?_ hg, ho'.integrableAtFilter ?_ hg'⟩, hf⟩
state_before: case refine_1
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁸ : MeasurableSpace α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
f : α → E
g : α → F
a b : α
μ : Measure α
l : Filter α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : CompactIccSpace α
g' : α → F
inst✝¹ : atBot.IsMeasurablyGenerated
inst✝ : atTop.IsMeasurablyGenerated
hf : LocallyIntegrable f μ
ho : f =O[atBot] g
hg : IntegrableAtFilter g atBot μ
ho' : f =O[atTop] g'
hg' : IntegrableAtFilter g' atTop μ
⊢ StronglyMeasurableAtFilter f atBot μ

case refine_2
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁸ : MeasurableSpace α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
f : α → E
g : α → F
a b : α
μ : Measure α
l : Filter α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : LinearOrder α
inst✝² : CompactIccSpace α
g' : α → F
inst✝¹ : atBot.IsMeasurablyGenerated
inst✝ : atTop.IsMeasurablyGenerated
hf : LocallyIntegrable f μ
ho : f =O[atBot] g
hg : IntegrableAtFilter g atBot μ
ho' : f =O[atTop] g'
hg' : IntegrableAtFilter g' atTop μ
⊢ StronglyMeasurableAtFilter f atTop μ
state_after: no goals
tactic: all_goals exact hf.aestronglyMeasurable.stronglyMeasurableAtFilter