I'm reading a note about optimal stopping in a Markovian setting
The author said that "a sequence $\left(X_{n}\right)_{n \in \mathbb{N}}$ of random variables with values in $\color{blue}{(E, \mathcal{E})}$". It seems to me that he meant a sequence $\left(X_{n}\right)_{n \in \mathbb{N}}$ of random variables with values in $\color{blue}{E}$, i.e., $$X_n: \Omega \to E$$
Could you verify if my understanding is correct?
Probably it means: values in $E$, measurable with respect to $\mathcal E$.
In order to be called a random variable, $f$ is supposed to be measurable. But unless the range set has an understood-algebra, it must be specified. In case the s et is $\mathbb R$ or $\mathbb C$, we automatically use the Borel sets.