Theorems · Theorem · measure theory

Antitone.measure_iInter

∀ {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : Preorder ι]
  [IsDirectedOrder ι] [Filter.atTop.IsCountablyGenerated] {s : ι → Set α},
  Antitone s → (∀ (i : ι), MeasureTheory.NullMeasurableSet (s i) μ) → (∃ i, μ (s i) ≠ ⊤) → μ (⋂ i, s i) = ⨅ i, μ (s i)

Continuity from above: the measure of the intersection of an antitone family of measurable sets indexed by a type with countably generated atTop filter is equal to the infimum of the measures.

Defined in

Mathlib.MeasureTheory.Measure.MeasureSpace

Cited by

2 results in Mathlib

Foundations

Depth 201 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PreorderIsDirectedOrderFilter.IsCountablyGenerated