Theorems · Theorem · measure theory

MeasureTheory.exists_subordinate_pairwise_disjoint

∀ {ι : Type u_1} {α : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] {s : ι → Set α},
  (∀ (i : ι), MeasureTheory.NullMeasurableSet (s i) μ) →
    Pairwise (Function.onFun (MeasureTheory.AEDisjoint μ) s) →
      ∃ t,
        (∀ (i : ι), t i ⊆ s i) ∧
          (∀ (i : ι), s i =ᵐ[μ] t i) ∧ (∀ (i : ι), MeasurableSet (t i)) ∧ Pairwise (Function.onFun Disjoint t)

If sᵢ is a countable family of (null) measurable pairwise μ-a.e. disjoint sets, then there exists a subordinate family tᵢ ⊆ sᵢ of measurable pairwise disjoint sets such that tᵢ =ᵐ[μ] sᵢ.

Defined in

Mathlib.MeasureTheory.Measure.NullMeasurable

Cited by

1 results in Mathlib

Foundations

Depth 184 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

Countable