Theorems · Theorem · measure theory

MeasureTheory.exists_Lp_half

∀ {α : Type u_1} (ε : Type u_3) {m : MeasurableSpace α} [inst : TopologicalSpace ε] [inst_1 : ESeminormedAddMonoid ε]
  (μ : MeasureTheory.Measure α) (p : ENNReal) {δ : ENNReal},
  δ ≠ 0 →
    ∃ η,
      0 < η ∧
        ∀ (f g : α → ε),
          MeasureTheory.AEStronglyMeasurable f μ →
            MeasureTheory.AEStronglyMeasurable g μ →
              MeasureTheory.eLpNorm f p μ ≤ η → MeasureTheory.eLpNorm g p μ ≤ η → MeasureTheory.eLpNorm (f + g) p μ < δ

Technical lemma to control the addition of functions in L^p even for p < 1: Given δ > 0, there exists η such that two functions bounded by η in L^p have a sum bounded by δ. One could take η = δ / 2 for p ≥ 1, but the point of the lemma is that it works also for p < 1.

Defined in

Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality

Cited by

4 results in Mathlib

Foundations

Depth 218 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceESeminormedAddMonoid