Theorems · Theorem · measure theory

MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_tendsto

∀ {ι : Type u_1} {E : Type u_2} {μ : MeasureTheory.Measure ℝ} {l : Filter ι} [l.NeBot] [l.IsCountablyGenerated]
  [inst : NormedAddCommGroup E] {a : ι → ℝ} {f : ℝ → E} (I b : ℝ),
  (∀ (i : ι), MeasureTheory.IntegrableOn f (Set.Ioc (a i) b) μ) →
    Filter.Tendsto a l Filter.atBot →
      Filter.Tendsto (fun i => ∫ (x : ℝ) in a i..b, ‖f x‖ ∂μ) l (nhds I) → MeasureTheory.IntegrableOn f (Set.Iic b) μ

If f is integrable on intervals Ioc (a i) b, where a i tends to -∞, and ∫ x in a i .. b, ‖f x‖ ∂μ converges to I : ℝ along a filter l, then f is integrable on the interval (-∞, b)

Defined in

Mathlib.MeasureTheory.Integral.IntegralEqImproper

Cited by

0 results in Mathlib

Foundations

Depth 256 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

Filter.NeBotFilter.IsCountablyGeneratedNormedAddCommGroup