Theorems · Theorem · measure theory

MeasureTheory.setIntegral_condExpL1CLM

∀ {α : Type u_1} {F' : Type u_3} [inst : NormedAddCommGroup F'] [inst_1 : NormedSpace ℝ F'] {m m0 : MeasurableSpace α}
  {μ : MeasureTheory.Measure α} {hm : m ≤ m0} [inst_2 : MeasureTheory.SigmaFinite (μ.trim hm)] {s : Set α}
  [inst_3 : CompleteSpace F'] (f : ↥(MeasureTheory.Lp F' 1 μ)),
  MeasurableSet s → ∫ (x : α) in s, ↑↑((MeasureTheory.condExpL1CLM F' hm μ) f) x ∂μ = ∫ (x : α) in s, ↑↑f x ∂μ

The integral of the conditional expectation condExpL1CLM over an m-measurable set is equal to the integral of f on that set. See also setIntegral_condExp, the similar statement for condExp.

Defined in

Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Cited by

1 results in Mathlib

Foundations

Depth 289 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceMeasureTheory.SigmaFiniteCompleteSpace