Theorems · Theorem · measure theory

MeasurableSet.exists_lt_isClosed_of_ne_top

∀ {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} [inst_1 : TopologicalSpace α]
  [μ.WeaklyRegular] ⦃A : Set α⦄, MeasurableSet A → μ A ≠ ⊤ → ∀ {r : ENNReal}, r < μ A → ∃ K ⊆ A, IsClosed K ∧ r < μ K

Given a weakly regular measure, any measurable set of finite mass can be approximated from inside by closed sets.

Defined in

Mathlib.MeasureTheory.Measure.Regular

Cited by

0 results in Mathlib

Foundations

Depth 198 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceTopologicalSpaceMeasureTheory.Measure.WeaklyRegular