Theorems · Theorem · measure theory

MeasureTheory.Lp.induction_stronglyMeasurable_aux

∀ {α : Type u_1} {F : Type u_2} {p : ENNReal} [inst : NormedAddCommGroup F] {m m0 : MeasurableSpace α}
  {μ : MeasureTheory.Measure α} [inst_1 : Fact (1 ≤ p)] [inst_2 : NormedSpace ℝ F] (hm : m ≤ m0),
  p ≠ ⊤ →
    ∀ (P : ↥(MeasureTheory.Lp F p μ) → Prop),
      (∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤),
          P ↑(MeasureTheory.Lp.simpleFunc.indicatorConst p ⋯ ⋯ c)) →
        (∀ ⦃f g : α → F⦄ (hf : MeasureTheory.MemLp f p μ) (hg : MeasureTheory.MemLp g p μ),
            MeasureTheory.AEStronglyMeasurable f μ →
              MeasureTheory.AEStronglyMeasurable g μ →
                Disjoint (Function.support f) (Function.support g) →
                  P (MeasureTheory.MemLp.toLp f hf) →
                    P (MeasureTheory.MemLp.toLp g hg) →
                      P (MeasureTheory.MemLp.toLp f hf + MeasureTheory.MemLp.toLp g hg)) →
          IsClosed {f | P ↑f} → ∀ (f : ↥(MeasureTheory.Lp F p μ)), MeasureTheory.AEStronglyMeasurable (↑↑f) μ → P f

Auxiliary lemma for Lp.induction_stronglyMeasurable.

Defined in

Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable

Cited by

1 results in Mathlib

Foundations

Depth 242 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupFactNormedSpace