Theorems · Theorem · measure theory

MeasureTheory.MemLp.eLpNorm_indicator_norm_ge_pos_le

∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : NormedAddCommGroup β]
  {p : ENNReal} {f : α → β},
  MeasureTheory.MemLp f p μ →
    MeasureTheory.StronglyMeasurable f →
      ∀ {ε : ℝ}, 0 < ε → ∃ M, 0 < M ∧ MeasureTheory.eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ ENNReal.ofReal ε

This lemma implies that a single function is uniformly integrable (in the probability sense).

Defined in

Mathlib.MeasureTheory.Function.UniformIntegrable

Cited by

2 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroup