Theorems · Theorem · measure theory

MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets

∀ {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) {β : Type u_2} [inst : CompleteLinearOrder β]
  [DenselyOrdered β] [inst_2 : TopologicalSpace β] [OrderTopology β] [SecondCountableTopology β]
  [inst_5 : MeasurableSpace β] [BorelSpace β] (s : Set β),
  s.Countable →
    Dense s →
      ∀ (f : α → β),
        (∀ p ∈ s,
            ∀ q ∈ s,
              p < q →
                ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧ {x | f x < p} ⊆ u ∧ {x | q < f x} ⊆ v ∧ μ (u ∩ v) = 0) →
          AEMeasurable f μ

If a function f : α → β is such that the level sets {f < p} and {q < f} have measurable supersets which are disjoint up to measure zero when p < q, then f is almost-everywhere measurable. It is even enough to have this for p and q in a countable dense set.

Defined in

Mathlib.MeasureTheory.Function.AEMeasurableOrder

Cited by

1 results in Mathlib

Foundations

Depth 172 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CompleteLinearOrderDenselyOrderedTopologicalSpaceOrderTopologySecondCountableTopologyMeasurableSpaceBorelSpace