Adapted — Mathlib

English

If u and v are adapted with respect to a filtration f, then their pointwise product u·v is adapted with respect to the same filtration. This holds for any i in the index set.

Русский

Если функции u и v адаптированы относительно фильтрации f, то их точечное произведение u·v адаптировано по той же фильтрации.

LaTeX

$$$$ \\forall i, \\ (hu(i)) \\cdot (hv(i)) \\in \\text{Adapted}(f, u\\cdot v) $$$$

Lean4

/-- A sequence of functions `u` is adapted to a filtration `f` if for all `i`,
`u i` is `f i`-measurable. -/
def Adapted (f : Filtration ι m) (u : ι → Ω → β) : Prop :=
  ∀ i : ι, StronglyMeasurable[f i] (u i)

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