Theorems · Theorem · measure theory

MeasureTheory.unifIntegrable_of_tendsto_Lp

∀ {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},
  1 ≤ p →
    p ≠ ⊤ →
      (∀ (n : ℕ), MeasureTheory.MemLp (f n) p μ) →
        MeasureTheory.MemLp g p μ →
          Filter.Tendsto (fun n => MeasureTheory.eLpNorm (f n - g) p μ) Filter.atTop (nhds 0) →
            MeasureTheory.UnifIntegrable f p μ

Convergence in Lp implies uniform integrability.

Defined in

Mathlib.MeasureTheory.Function.UniformIntegrable

Cited by

2 results in Mathlib

Foundations

Depth 225 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroup