Theorems · Theorem · measure theory

MeasurableSet.measure_eq_iSup_isClosed_of_ne_top

∀ {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} [inst_1 : TopologicalSpace α]
  [μ.WeaklyRegular] ⦃A : Set α⦄, MeasurableSet A → μ A ≠ ⊤ → μ A = ⨆ K, ⨆ (_ : K ⊆ A), ⨆ (_ : IsClosed K), μ K

Given a weakly regular measure, any measurable set of finite mass can be approximated from inside by closed sets.

Defined in

Mathlib.MeasureTheory.Measure.Regular

Cited by

0 results in Mathlib

Foundations

Depth 198 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceTopologicalSpaceMeasureTheory.Measure.WeaklyRegular