Theorems · Theorem · measure theory

Directed.measure_iInter

∀ {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] {s : ι → Set α},
  (∀ (i : ι), MeasureTheory.NullMeasurableSet (s i) μ) →
    Directed (fun x1 x2 => x1 ⊇ x2) s → (∃ i, μ (s i) ≠ ⊤) → μ (⋂ i, s i) = ⨅ i, μ (s i)

Continuity from above: the measure of the intersection of a directed downwards countable family of measurable sets is the infimum of the measures.

Defined in

Mathlib.MeasureTheory.Measure.MeasureSpace

Cited by

1 results in Mathlib

Foundations

Depth 199 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

Countable