Theorems · Definition · probability

MeasureTheory.Filtration.natural

{Ω : Type u_1} →
  {ι : Type u_2} →
    {m : MeasurableSpace Ω} →
      {β : ι → Type u_3} →
        [inst : (i : ι) → TopologicalSpace (β i)] →
          [∀ (i : ι), TopologicalSpace.MetrizableSpace (β i)] →
            [mβ : (i : ι) → MeasurableSpace (β i)] →
              [∀ (i : ι), BorelSpace (β i)] →
                [inst_3 : Preorder ι] →
                  (u : (i : ι) → Ω → β i) →
                    (∀ (i : ι), MeasureTheory.StronglyMeasurable (u i)) → MeasureTheory.Filtration ι m

Given a sequence of functions, the natural filtration is the smallest sequence of σ-algebras such that the sequence of functions is measurable with respect to the filtration.

Defined in

Mathlib.Probability.Process.Filtration

Cited by

6 results in Mathlib

Foundations

Depth 165 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpace.MetrizableSpaceMeasurableSpaceBorelSpacePreorder