Theorems · Theorem · measure theory

MeasurableSet.exists_isCompact_diff_lt

Deprecated since 2026-06-03Use MeasurableSet.exists_isCompact_sdiff_lt instead.

∀ {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} [inst_1 : TopologicalSpace α]
  [OpensMeasurableSpace α] [T2Space α] [μ.InnerRegularCompactLTTop] ⦃A : Set α⦄,
  MeasurableSet A → μ A ≠ ⊤ → ∀ {ε : ENNReal}, ε ≠ 0 → ∃ K ⊆ A, IsCompact K ∧ μ (A \ K) < ε

Alias of MeasurableSet.exists_isCompact_sdiff_lt. 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 and MeasurableSet.exists_lt_isCompact_of_ne_top.

Defined in

Mathlib.MeasureTheory.Measure.Regular

Cited by

0 results in Mathlib

Foundations

Depth 196 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceTopologicalSpaceOpensMeasurableSpaceT2SpaceMeasureTheory.Measure.InnerRegularCompactLTTop