Let $\{M^{(n)}\}$ a Cauchy sequence in $X_1$. Then $\sup_{0\leq t\leq T}|M_t^{(n)}-M_t^{(m)}|\to 0$ when $m,n\to +\infty$ in $L^2(P)$. In particular, for each $t$ fixed in $[0,T]$, the sequence $\{M_t^{(n)}\}$ is Cauchy in $L^2(P)$. Since this space is complete, we can define $M_t$ as the limit in $L^2(P)$ of this sequence. Now, have have several things to check:
- We have that for all $\varepsilon >0$, we can find $n_0$ such that if $m,n\geq n_0$ and $t\in [0,T]$ then $\lVert M_t^{(n)}-M_t^{(m)}\rVert_{L^2(P)}\leq \varepsilon$ (since $|M_t^{(n)}-M_t^{(m)}|\leq \sup_{0\leq s\leq T}|M_s^{(n)}-M_s^{(m)}|$). Taking the limit $m\to +\infty$, we get that $\lVert M-M^{(n)}\rVert_{X_1}\to 0$. Indeed, you can check, using the fact that converge in $L^2$ implies convergence of a subsequence almost everywhere, that for each subsequence it's true for a further subsequence.
- We check that $M\in X_1$. We look at right continuity. We can find an increasing sequence $\{n_k\}$ of integers such that $\sup_{0\leq t\leq T}|M^{(n_k)}_t-M_t|\to 0$ almost everywhere (say in $\Omega'$. Fix $t_0\in [0,T]$. For $s\geq t_0$ and $\omega\in\Omega'$, we have \begin{align} |M_s(\omega)-M_{t_0}(\omega)|&\leq |M_s(\omega)-M_s^{(n_k)}(\omega)|\\ &+|M_s^{(n_k)}(\omega)-M_{t_0}^{(n_k)}(\omega)|+|M_{t_0}^{(n_k)}(\omega)-M_{t_0}(\omega)|\\ &\leq 2\sup_{0\leq t\leq T}|M_t(\omega)-M_t^{(n_k)}(\omega)|+|M_s^{(n_k)}(\omega)-M_{t_0}^{(n_k)}(\omega)|. \end{align} Now, we fix $\varepsilon>0$, and pick $k$ such that $2\sup_{0\leq t\leq T}|M_t(\omega)-M_t^{(n_k)}(\omega)|\leq \varepsilon$. Then we conclude using the right continuity of $M^{(n_k)}$. We have now to see left limit. It follows a similar argument, plus the Cauchy criterion: fix $t_0\in [0,T]$. For $s_1,s_2\leq t$ we have $|M_{s_1}(\omega)-M_{s_2}(\omega)|\leq 2\sup_{0\leq t\leq T}|M_t(\omega)-M_t^{(n_k)}(\omega)|+|M_{s_1}^{(n_k)}(\omega)-M_{s_2}^{(n_k)}(\omega)|.$
- We have to check that $\{M_t\}$ is adapted. To see that, note that the set $\{\omega,\{M_t^{(n_k)}(\omega)\}\mbox{ doesn't converge}\}$ can we written as a countable intersection of countable union of elements of $\mathcal F_t$.