Theorems · Theorem · measure theory

MeasurableSet.exists_isCompact_isClosed_sdiff_lt

∀ {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} [inst_1 : TopologicalSpace α] [BorelSpace α]
  [R1Space α] [μ.InnerRegularCompactLTTop] ⦃A : Set α⦄,
  MeasurableSet A → μ A ≠ ⊤ → ∀ {ε : ENNReal}, ε ≠ 0 → ∃ K ⊆ A, IsCompact K ∧ IsClosed K ∧ μ (A \ K) < ε

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 closed subset. Compared to MeasurableSet.exists_isCompact_sdiff_lt, this lemma additionally assumes that α is an R₁ space with Borel σ-algebra.

Defined in

Mathlib.MeasureTheory.Measure.Regular

Cited by

5 results in Mathlib

Foundations

Depth 195 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceTopologicalSpaceBorelSpaceR1SpaceMeasureTheory.Measure.InnerRegularCompactLTTop