Theorems · Theorem · measure theory

MeasureTheory.measure_iInter_of_ae_monotone

∀ {α : Type u_1} {ι : Type u_5} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : Preorder ι]
  [IsCodirectedOrder ι] [Filter.atBot.IsCountablyGenerated] {s : ι → Set α},
  (∀ᵐ (ω : α) ∂μ, Monotone fun x => ω ∈ s x) →
    (∀ (i : ι), MeasureTheory.NullMeasurableSet (s i) μ) → (∃ i, μ (s i) ≠ ⊤) → μ (⋂ i, s i) = ⨅ i, μ (s i)

Continuity from above (a.e. version): the measure of the intersection of a family of sets that is almost everywhere monotone is equal to the infimum of the measures.

Defined in

Mathlib.MeasureTheory.Measure.MeasureSpace

Cited by

1 results in Mathlib

Foundations

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

Assumes

PreorderIsCodirectedOrderFilter.IsCountablyGenerated