What does the following "fraction" mean in the context of: for every $\mathcal{F}_s^{t,0} \big{/} B(\mathbb{R}^n)$ measurable random variable.
When we work with $$\mu (\Omega , \mathcal{F} , \mathcal {F}_s^t , \mathbb{P} , W)$$ where $\Omega , \mathcal{F} , \mathbb{P}$ is a complete probability space, $\mathcal{F}^t_s$ a right continuous complete filtration and a natural filtration to be $F_s^{t,0} = \pmb{\sigma}(W(r) : t \le r \le s)$ where here $\pmb{\sigma}(A)$ is the sigma algebra generated by $A$. We also have $W$ a Wiener process such that $W(t_2) - W(t_1) \perp \mathcal{F}_{t_1}^t , t_2 > t_1$, $W(t_2) - W(t_1) \sim N(0 , (t_2 - t_1)I)$ and $W$ has continuous trajectories in $\mathbb{P}-a.s.$. $B$ is borel.
Let $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ be two measurable spaces (i.e. $\mathcal{A}$ is a $\sigma$-algebra on $X$ and $\mathcal{B}$ a $\sigma$-algebra on $Y$). A mapping $f:X \to Y$ is $\mathcal{A}/\mathcal{B}$-measurable if $f^{-1}(B) \in \mathcal{A}$ for all $B \in \mathcal{B}$. This can be written more compactly as $$f^{-1}(\mathcal{B}) \subseteq \mathcal{A}.$$
In your setting, $\mathcal{B}:=\mathcal{B}(\mathbb{R}^n)$ is the Borel $\sigma$-algebra on $\mathbb{R}^n$ and $\mathcal{A}:=\mathcal{F}_s^{t,0}$ is the canonical filtration associated with the process $(W_{t+s})_{s \geq 0}$. Consequently, a mapping $f: \Omega \to \mathbb{R}^n$ is $\mathcal{F}_s^{t,0}/\mathcal{B}(\mathbb{R}^n)$-measurable if $$f^{-1}(B) \in \mathcal{F}_{s}^{t,0} \quad \text{for all $B \in \mathcal{B}(\mathbb{R}^n)$}.$$