Theorems · Theorem · measure theory

MeasurableSet.exists_isOpen_symmDiff_lt

∀ {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} [inst_1 : TopologicalSpace α]
  [μ.InnerRegularCompactLTTop] [MeasureTheory.IsLocallyFiniteMeasure μ] [R1Space α] [BorelSpace α] {s : Set α},
  MeasurableSet s → μ s ≠ ⊤ → ∀ {ε : ENNReal}, ε ≠ 0 → ∃ U, IsOpen U ∧ μ U < ⊤ ∧ μ (symmDiff U s) < ε

Let μ be a locally finite measure on an R₁ topological space with Borel σ-algebra. If μ is inner regular for finite measure sets with respect to compact sets, then any measurable set of finite measure can be approximated in measure by an open set. See also Set.exists_isOpen_lt_of_lt and MeasurableSet.exists_isOpen_diff_lt for the case of an outer regular measure.

Defined in

Mathlib.MeasureTheory.Measure.Regular

Cited by

1 results in Mathlib

Foundations

Depth 202 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceTopologicalSpaceMeasureTheory.Measure.InnerRegularCompactLTTopMeasureTheory.IsLocallyFiniteMeasureR1SpaceBorelSpace