Theorems · Theorem · measure theory

MeasurableSet.exists_lt_isCompact_of_ne_top

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

If μ is inner regular for finite measure sets with respect to compact sets, then any measurable set of finite measure can be approximated by a compact subset. See also MeasurableSet.exists_isCompact_lt_add.

Defined in

Mathlib.MeasureTheory.Measure.Regular

Cited by

4 results in Mathlib

Foundations

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

Assumes

MeasurableSpaceTopologicalSpaceMeasureTheory.Measure.InnerRegularCompactLTTop