Theorems · Theorem · measure theory

MeasureTheory.MemLp.induction_dense

∀ {α : Type u_1} {E : Type u_4} [inst : MeasurableSpace α] [inst_1 : NormedAddCommGroup E] {p : ENNReal}
  {μ : MeasureTheory.Measure α},
  p ≠ ⊤ →
    ∀ (P : (α → E) → Prop),
      (∀ (c : E) ⦃s : Set α⦄,
          MeasurableSet s →
            μ s < ⊤ → ∀ {ε : ENNReal}, ε ≠ 0 → ∃ g, MeasureTheory.eLpNorm (g - s.indicator fun x => c) p μ ≤ ε ∧ P g) →
        (∀ (f g : α → E), P f → P g → P (f + g)) →
          (∀ (f : α → E), P f → MeasureTheory.AEStronglyMeasurable f μ) →
            ∀ {f : α → E},
              MeasureTheory.MemLp f p μ → ∀ {ε : ENNReal}, ε ≠ 0 → ∃ g, MeasureTheory.eLpNorm (f - g) p μ ≤ ε ∧ P g

If a set of ae strongly measurable functions is stable under addition and approximates characteristic functions in ℒp, then it is dense in ℒp.

Defined in

Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Cited by

2 results in Mathlib

Foundations

Depth 222 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceNormedAddCommGroup