Theorems · Theorem · measure theory

MeasurableSet.exists_isClosed_lt_add

∀ {α : Type u_1} [inst : MeasurableSpace α] {μ : MeasureTheory.Measure α} [inst_1 : TopologicalSpace α]
  [μ.WeaklyRegular] {s : Set α},
  MeasurableSet s → μ s ≠ ⊤ → ∀ {ε : ENNReal}, ε ≠ 0 → ∃ K ⊆ s, IsClosed K ∧ μ s < μ K + ε

If s is a measurable set, a weakly regular measure μ is finite on s, and ε is a positive number, then there exist a closed set K ⊆ s such that μ s < μ K + ε.

Defined in

Mathlib.MeasureTheory.Measure.Regular

Cited by

2 results in Mathlib

Foundations

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

Assumes

MeasurableSpaceTopologicalSpaceMeasureTheory.Measure.WeaklyRegular