Theorems · Theorem · measure theory

Monotone.measure_iUnion

∀ {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}, Monotone s → μ (⋃ i, s i) = ⨆ i, μ (s i)

Continuity from below: the measure of the union of a monotone family of sets is equal to the supremum of their measures. The theorem assumes that the atTop filter on the index set is countably generated, so it works for a family indexed by a countable type, as well as ℝ.

Defined in

Mathlib.MeasureTheory.Measure.MeasureSpace

Cited by

10 results in Mathlib

Foundations

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

Assumes

PreorderIsDirectedOrderFilter.IsCountablyGenerated