Theorems · Theorem · measure theory

IsNowhereDense.of_isClosed_null

∀ {X : Type u_1} [inst : TopologicalSpace X] [inst_1 : MeasurableSpace X] {s : Set X} {μ : MeasureTheory.Measure X}
  [μ.IsOpenPosMeasure], IsClosed s → μ s = 0 → IsNowhereDense s

A closed measure zero subset is nowhere dense. (Closedness is required: for instance, the rational numbers are countable (thus have measure zero), but are dense (hence not nowhere dense).)

Defined in

Mathlib.MeasureTheory.Measure.OpenPos

Cited by

1 results in Mathlib

Foundations

Depth 177 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceMeasurableSpaceMeasureTheory.Measure.IsOpenPosMeasure