tail field

definition A tail field is a way to look at events that that happen arbitrarily far down an infinite sequence of events.

Definition

For a probability space $(\Omega ,\mathcal{F},P)$, let ${\mathcal{F}}_1,{\mathcal{F}}_2,\cdots \subset \mathcal{F}$ be a sequence of sigma algebras generated by events. The tail field is the 20231205121417-sigma_algebra

$\mathcal{T}={\bigcap }_{m=1}^{\infty }\sigma \left({\bigcup }_{n=m}^{\infty }{\mathcal{F}}_n\right)$