Embedding of random vectors into continuous martingales

Let E be a real, separable Banach space and denote by ${\displaystyle L^0(\Omega ,E)}$ the space of all E-valued random vectors defined on the probability space Ω. The following result is proved. There exists an extension ${\displaystyle \left\{w\tilde{\Omega }\right\}}$ of Ω, and a filtration ${\displaystyle {\left({\left\{w\widetilde{ℱ}\right\}}_t\right)}_{t\ge 0}}$ on ${\displaystyle \left\{w\tilde{\Omega }\right\}}$, such that for every ${\displaystyle X\in L^0(\Omega ,E)}$ there is an E-valued, continuous ${\displaystyle \left({\left\{w\widetilde{ℱ}\right\}}_t\right)}$-martingale ${\displaystyle {\left(M_t\left(X\right)\right)}_{t\ge 0}}$ in which X is embedded in the sense that ${\displaystyle X=M_{\tau }\left(X\right)}$ a.s. for an a.s. finite stopping time τ. For E = ℝ this gives a Skorokhod embedding for all ${\displaystyle X\in L^0(\Omega ,ℝ)}$, and for general E this leads to a representation of random vectors as stochastic integrals relative to a Brownian motion.