File: Mathlib/MeasureTheory/Integral/Asymptotics.lean

full_name: Asymptotics.IsBigO.integrableAtFilter
code: /-- If `f = O[l] g` on measurably generated `l`, `f` is strongly measurable at `l`,
and `g` is integrable at `l`, then `f` is integrable at `l`. -/
theorem _root_.Asymptotics.IsBigO.integrableAtFilter [IsMeasurablyGenerated l]
    (hf : f =O[l] g) (hfm : StronglyMeasurableAtFilter f l μ) (hg : IntegrableAtFilter g l μ) :
    IntegrableAtFilter f l μ 
goal: MeasureTheory.IntegrableAtFilter f l μ

full_name: Asymptotics.IsBigO.integrable
code: /-- Variant of `MeasureTheory.Integrable.mono` taking `f =O[⊤] (g)` instead of `‖f(x)‖ ≤ ‖g(x)‖` -/
theorem _root_.Asymptotics.IsBigO.integrable (hfm : AEStronglyMeasurable f μ)
    (hf : f =O[⊤] g) (hg : Integrable g μ) : Integrable f μ 
goal: MeasureTheory.Integrable f μ

full_name: MeasureTheory.LocallyIntegrable.integrable_of_isBigO_cocompact
code: /-- If `f` is locally integrable, and `f =O[cocompact] g` for some `g` integrable at `cocompact`,
then `f` is integrable. -/
theorem LocallyIntegrable.integrable_of_isBigO_cocompact [IsMeasurablyGenerated (cocompact α)]
    (hf : LocallyIntegrable f μ) (ho : f =O[cocompact α] g)
    (hg : IntegrableAtFilter g (cocompact α) μ) : Integrable f μ 
goal: MeasureTheory.Integrable f μ

full_name: MeasureTheory.LocallyIntegrable.integrable_of_isBigO_atBot_atTop
code: /-- If `f` is locally integrable, and `f =O[atBot] g`, `f =O[atTop] g'` for some
`g`, `g'` integrable at `atBot` and `atTop` respectively, then `f` is integrable. -/
theorem LocallyIntegrable.integrable_of_isBigO_atBot_atTop
    [IsMeasurablyGenerated (atBot (α := α))] [IsMeasurablyGenerated (atTop (α := α))]
    (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 
goal: MeasureTheory.Integrable f μ

full_name: MeasureTheory.LocallyIntegrableOn.integrableOn_of_isBigO_atBot
code: /-- If `f` is locally integrable on `(∞, a]`, and `f =O[atBot] g`, for some
`g` integrable at `atBot`, then `f` is integrable on `(∞, a]`. -/
theorem LocallyIntegrableOn.integrableOn_of_isBigO_atBot [IsMeasurablyGenerated (atBot (α := α))]
    (hf : LocallyIntegrableOn f (Iic a) μ) (ho : f =O[atBot] g)
    (hg : IntegrableAtFilter g atBot μ) : IntegrableOn f (Iic a) μ 
goal: MeasureTheory.IntegrableOn f (Set.Iic a) μ

full_name: MeasureTheory.LocallyIntegrableOn.integrableOn_of_isBigO_atTop
code: /-- If `f` is locally integrable on `[a, ∞)`, and `f =O[atTop] g`, for some
`g` integrable at `atTop`, then `f` is integrable on `[a, ∞)`. -/
theorem LocallyIntegrableOn.integrableOn_of_isBigO_atTop [IsMeasurablyGenerated (atTop (α := α))]
    (hf : LocallyIntegrableOn f (Ici a) μ) (ho : f =O[atTop] g)
    (hg : IntegrableAtFilter g atTop μ) : IntegrableOn f (Ici a) μ 
goal: MeasureTheory.IntegrableOn f (Set.Ici a) μ

full_name: MeasureTheory.LocallyIntegrable.integrable_of_isBigO_atBot
code: /-- If `f` is locally integrable, `f` has a top element, and `f =O[atBot] g`, for some
`g` integrable at `atBot`, then `f` is integrable. -/
theorem LocallyIntegrable.integrable_of_isBigO_atBot [IsMeasurablyGenerated (atBot (α := α))]
    [OrderTop α] (hf : LocallyIntegrable f μ) (ho : f =O[atBot] g)
    (hg : IntegrableAtFilter g atBot μ) : Integrable f μ 
goal: MeasureTheory.Integrable f μ

full_name: MeasureTheory.LocallyIntegrable.integrable_of_isBigO_atTop
code: /-- If `f` is locally integrable, `f` has a bottom element, and `f =O[atTop] g`, for some
`g` integrable at `atTop`, then `f` is integrable. -/
theorem LocallyIntegrable.integrable_of_isBigO_atTop [IsMeasurablyGenerated (atTop (α := α))]
    [OrderBot α] (hf : LocallyIntegrable f μ) (ho : f =O[atTop] g)
    (hg : IntegrableAtFilter g atTop μ) : Integrable f μ 
goal: MeasureTheory.Integrable f μ

full_name: MeasureTheory.LocallyIntegrable.integrable_of_isBigO_atTop_of_norm_isNegInvariant
code: /-- If `f` is locally integrable, `‖f(-x)‖ = ‖f(x)‖`, and `f =O[atTop] g`, for some
`g` integrable at `atTop`, then `f` is integrable. -/
theorem LocallyIntegrable.integrable_of_isBigO_atTop_of_norm_isNegInvariant
    [IsMeasurablyGenerated (atTop (α := α))] [MeasurableNeg α] [μ.IsNegInvariant]
    (hf : LocallyIntegrable f μ) (hsymm : norm ∘ f =ᵐ[μ] norm ∘ f ∘ Neg.neg) (ho : f =O[atTop] g)
    (hg : IntegrableAtFilter g atTop μ) : Integrable f μ 
goal: MeasureTheory.Integrable f μ