Theorems · Theorem · measure theory

aestronglyMeasurable_iff_aemeasurable_separable

∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace β] {m₀ : MeasurableSpace α} {μ : MeasureTheory.Measure α}
  {f : α → β} [TopologicalSpace.PseudoMetrizableSpace β] [inst_2 : MeasurableSpace β] [BorelSpace β],
  MeasureTheory.AEStronglyMeasurable f μ ↔
    AEMeasurable f μ ∧ ∃ t, TopologicalSpace.IsSeparable t ∧ ∀ᵐ (x : α) ∂μ, f x ∈ t

A function is almost everywhere strongly measurable if and only if it is almost everywhere measurable, and up to a zero measure set its range is contained in a separable set.

Defined in

Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Cited by

10 results in Mathlib

Foundations

Depth 200 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpace.PseudoMetrizableSpaceMeasurableSpaceBorelSpace