Consider the probability space $(S,\mathcal{A}, P)$. Let $Y$ be a $(\mathcal{Y}, \mathcal{C}_{\mathcal{Y}})$-valued random variable, where $\mathcal{Y}$ is the space of all Borel measurable functions from $[0,1]$ to $\{0,1\}$. In other words, $Y(s)$ is a Borel measurable function from $[0,1]$ to $\{0,1\}$.
Is this equivalent to define a stochastic process, i.e. a collection of $(\{0,1\}, \mathcal{B}_{\{0,1\}})$-valued random variables
$$ \{X_t \text{ s.t. } t \in [0,1]\} $$
with $X_t(s)\equiv Y(s)(t)$?
Do we have any restriction on what should be $\mathcal{C}_{\mathcal{Y}}$?