Theorems · Theorem · measure theory

MeasureTheory.Measure.MeasureDense.fin_meas_approx

∀ {X : Type u_1} [m : MeasurableSpace X] {μ : MeasureTheory.Measure X} {𝒜 : Set (Set X)},
  μ.MeasureDense 𝒜 →
    ∀ {s : Set X}, MeasurableSet s → μ s ≠ ⊤ → ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, μ t ≠ ⊤ ∧ μ (symmDiff s t) < ENNReal.ofReal ε

If a family of sets 𝒜 is measure-dense in X, then any measurable set with finite measure can be approximated by sets in 𝒜 with finite measure.

Defined in

Mathlib.MeasureTheory.Measure.SeparableMeasure

Cited by

2 results in Mathlib

Foundations

Depth 174 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpace