Theorems · Theorem · measure theory

MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm_of_ne_top

∀ {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 : SeminormedAddCommGroup E],
  p ≠ 0 →
    p ≠ ⊤ →
      (∀ (n : ι), MeasureTheory.AEStronglyMeasurable (f n) μ) →
        MeasureTheory.AEStronglyMeasurable g μ →
          ∀ {l : Filter ι},
            Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) p μ) l (nhds 0) →
              MeasureTheory.TendstoInMeasure μ f l g

This lemma is superseded by MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm where we allow p = ∞.

Defined in

Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Cited by

1 results in Mathlib

Foundations

Depth 211 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SeminormedAddCommGroup