Theorems · Theorem · measure theory

MeasureTheory.setLIntegral_nnnorm_condExpL2_indicator_le

∀ {α : Type u_1} {E' : Type u_3} {𝕜 : Type u_7} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E']
  [inst_2 : InnerProductSpace 𝕜 E'] [inst_3 : CompleteSpace E'] [NormedSpace ℝ E'] {m m0 : MeasurableSpace α}
  {μ : MeasureTheory.Measure α} {s : Set α} (hm : m ≤ m0) (hs : MeasurableSet s) (hμs : μ s ≠ ⊤) (x : E') {t : Set α},
  MeasurableSet t →
    μ t ≠ ⊤ →
      ∫⁻ (a : α) in t, ↑‖↑↑↑((MeasureTheory.condExpL2 E' 𝕜 hm) (MeasureTheory.indicatorConstLp 2 hs hμs x)) a‖₊ ∂μ ≤
        μ (s ∩ t) * ↑‖x‖₊

Defined in

Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Cited by

2 results in Mathlib

Foundations

Depth 276 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceCompleteSpaceNormedSpace