Constant Function is a Stopping Time

Last updated Nov 1, 2022

Let $\mathcal{F} = {\mathcal{B}_{n} : n \in \mathbb{N}}$ be a Discrete-Time Filtration over $\Omega$, $k \in \bar{\mathbb{N}}$, and let $\nu : \Omega \to \bar{\mathbb{N}}$ be defined as $\nu(\omega) = k$ for all $\omega \in \Omega$. Then $\nu$ is a Stopping Time.

We simply verify the definition of a Stopping Time. Let $n \in \mathbb{N}$. Observe that $$[\nu = n] = \begin{cases} \emptyset &\text{if }n \neq k\\ \Omega &\text{otherwise}\end{cases}$$ All Sigma Algebras on $\Omega$ contain $\Omega$ and $\emptyset$ (by definition), so certainly $[\nu = n] \in \mathcal{B}_{n}$. $\blacksquare$