Theorems · Theorem · general topology

Measurable.exists_continuous

∀ {α : Type u_3} {β : Type u_4} [t : TopologicalSpace α] [PolishSpace α] [inst : MeasurableSpace α] [BorelSpace α]
  [tβ : TopologicalSpace β] [inst_2 : MeasurableSpace β] [OpensMeasurableSpace β] {f : α → β}
  [SecondCountableTopology ↑(Set.range f)], Measurable f → ∃ t' ≤ t, Continuous f ∧ PolishSpace α

Given a Borel-measurable function from a Polish space to a second-countable space, there exists a finer Polish topology on the source space for which the function is continuous.

Defined in

Mathlib.MeasureTheory.Constructions.Polish.Basic

Cited by

2 results in Mathlib

Foundations

Depth 169 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpacePolishSpaceMeasurableSpaceBorelSpaceTopologicalSpaceMeasurableSpaceOpensMeasurableSpaceSecondCountableTopology