Theorems · Theorem · measure theory

MeasureTheory.tendstoInMeasure_iff_tendsto_Lp_finite

∀ {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.MemLp (f n) p μ) →
        MeasureTheory.MemLp g p μ →
          (MeasureTheory.TendstoInMeasure μ f Filter.atTop g ∧ MeasureTheory.UnifIntegrable f p μ ↔
            Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) p μ) Filter.atTop (nhds 0))

Vitali's convergence theorem: A sequence of functions f converges to g in Lp if and only if it is uniformly integrable and converges to g in measure.

Defined in

Mathlib.MeasureTheory.Function.UniformIntegrable

Cited by

0 results in Mathlib

Foundations

Depth 226 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupMeasureTheory.IsFiniteMeasure