Theorems · Theorem · measure theory

MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm

∀ {α : Type u_1} {ι : Type u_2} {E : Type u_4} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p : ENNReal}
  {f : ι → α → E} {g : α → E} [inst : NormedAddCommGroup E] {l : Filter ι},
  p ≠ 0 →
    (∀ (n : ι), MeasureTheory.AEStronglyMeasurable (f n) μ) →
      MeasureTheory.AEStronglyMeasurable g μ →
        Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) p μ) l (nhds 0) →
          MeasureTheory.TendstoInMeasure μ f l g

Convergence in Lp implies convergence in measure.

Defined in

Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Cited by

3 results in Mathlib

Foundations

Depth 212 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroup