Theorems · Theorem · measure theory

MeasureTheory.measure_iInter_eq_iInf_measure_iInter_le

∀ {α : Type u_8} {ι : Type u_9} {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] [inst : Preorder ι]
  [IsDirectedOrder ι] {f : ι → Set α},
  (∀ (i : ι), MeasureTheory.NullMeasurableSet (f i) μ) →
    (∃ i, μ (f i) ≠ ⊤) → μ (⋂ i, f i) = ⨅ i, μ (⋂ j, ⋂ (_ : j ≤ i), f j)

Continuity from above: the measure of the intersection of a sequence of measurable sets is the infimum of the measures of the partial intersections.

Defined in

Mathlib.MeasureTheory.Measure.MeasureSpace

Cited by

1 results in Mathlib

Foundations

Depth 202 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CountablePreorderIsDirectedOrder