Theorems · Theorem · measure theory

MeasureTheory.tendsto_Lp_finite_of_tendstoInMeasure

∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : NormedAddCommGroup β]
  {p : ENNReal} {f : ℕ → α → β} {g : α → β} [MeasureTheory.IsFiniteMeasure μ],
  1 ≤ p →
    p ≠ ⊤ →
      (∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (f n) μ) →
        MeasureTheory.MemLp g p μ →
          MeasureTheory.UnifIntegrable f p μ →
            MeasureTheory.TendstoInMeasure μ f Filter.atTop g →
              Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) p μ) Filter.atTop (nhds 0)

Forward direction of Vitali's convergence theorem: if f is a sequence of uniformly integrable functions that converge in measure to some function g in a finite measure space, then f converge in Lp to g.

Defined in

Mathlib.MeasureTheory.Function.UniformIntegrable

Cited by

4 results in Mathlib

Foundations

Depth 224 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupMeasureTheory.IsFiniteMeasure