Let $(\Omega,\mathcal A,\operatorname P)$ be a probaiblity space, $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$ and $E$ be a normed $\mathbb R$-vector space. If $t\ge0$ and $(X_s)_{s\in[0,\:t]}$ is an $E$-valued $(\mathcal F_s)_{s\in[0,\:t]}$-adapted càdlàg process on $(\Omega,\mathcal A,\operatorname P)$, is $$\left\|X\right\|_t^\ast:=\sup_{s\in[0,\:t]}\left\|X_s\right\|_E$$ $\mathcal F_t$-measurable?
Moreover, I would like to know whether the space $\mathcal C$ of all such processes which additionally satisfy $$\left\|X\right\|_t:=\left\|\left\|X\right\|_t^\ast\right\|_{L^2(\operatorname P)}<\infty\tag1$$ equipped with $\|\;\cdot\;\|_t$ is complete, whenever $E$ is complete.
Do we really need to carry out the usual proof $L^2$-completeness, or can we somehow use that we already know that $L^2(\operatorname P)$ is complete? If the answer to my first question is positive, then every Cauchy sequence $(X^n)_{n\in\mathbb N}\subseteq\mathcal C$ is also a Cauchy sequence $\left(\left\|X^n\right\|_t^\ast\right)_{n\in\mathbb N}$ in $L^2(\operatorname P)$ ...