Theorems · Theorem · measure theory

Directed.measure_iUnion

∀ {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] {s : ι → Set α},
  Directed (fun x1 x2 => x1 ⊆ x2) s → μ (⋃ i, s i) = ⨆ i, μ (s i)

Continuity from below: the measure of the union of a directed sequence of (not necessarily measurable) sets is the supremum of the measures.

Defined in

Mathlib.MeasureTheory.Measure.MeasureSpace

Cited by

5 results in Mathlib

Foundations

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

Assumes

Countable