Theorems · Theorem · measure theory

MeasurableSet.exists_isCompact_isClosed_lt_add

∀ {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} [inst_1 : TopologicalSpace α]
  [μ.InnerRegularCompactLTTop] [R1Space α] [BorelSpace α] ⦃A : Set α⦄,
  MeasurableSet A → μ A ≠ ⊤ → ∀ {ε : ENNReal}, ε ≠ 0 → ∃ K ⊆ A, IsCompact K ∧ IsClosed K ∧ μ A < μ 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_lt_add, this version additionally assumes that α is an R₁ space with Borel σ-algebra.

Defined in

Mathlib.MeasureTheory.Measure.Regular

Cited by

1 results in Mathlib

Foundations

Depth 181 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceTopologicalSpaceMeasureTheory.Measure.InnerRegularCompactLTTopR1SpaceBorelSpace