Can someone translate $s_j : \Omega \hspace{3pt} x \hspace{3pt} [0,T_{Final}] \rightarrow S_j \subset R$ into simple English for me?

$S_j$ is a subset of $\mathbb{R}$.

And $s_j$ is a function from $\Omega \times [0,T_\mathrm{final})$ to $S_j$, i.e. $s_j$ takes a pair $(x,t)$ where $x$ belongs to $\Omega$ and $t$ belongs to $[0,T_\mathrm{final})$, and returns the value $s_j (x,t)$ which is a real number in the set $S_j$.