Theorems · Theorem · measure theory

MeasurableSet.exists_lt_isCompact

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

If μ is inner regular, then any measurable set can be approximated by a compact subset. See also MeasurableSet.exists_isCompact_lt_add_of_ne_top.

Defined in

Mathlib.MeasureTheory.Measure.Regular

Cited by

2 results in Mathlib

Foundations

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

Assumes

MeasurableSpaceTopologicalSpaceMeasureTheory.Measure.InnerRegular