Suppose that $(\Omega,\mathscr{F},P)$ is a complete probability space equipped a filtration $\{\mathscr{F}_t\}$ satisfying the usual conditions. $B_t$ is a 1-dimentional Brownian motion with respect to the filtration $\{\mathscr{F}_t\}$. Consider the 1-dimentional SDE

\begin{equation}dx(t)=f(x(t),t)dt+g(x(t),t)dB(t),\ (*)\end{equation}

with the initial value $x(t_0)=\xi\in L^2(\Omega;\mathbb{R})$, which statisfies the Lipschitz condition and the Linear growth condition.

Let \begin{equation}\Omega_0=\{\omega,\ there\ exist\ at\ least\ two\ different\ sample\ paths\ x_{\alpha}(t,\omega), x_{\beta}(t,\omega)\}\end{equation} Where $x_{\alpha}(t)$, $x_{\beta}(t)$ are both solutions of $(*)$.

By the existence and uniqueness theorem, $(*)$ has the pathwise unique solution, which means that if there are two solutions $x(t)$ and $\overline{x}(t)$, then $P\{x(t)=\overline{x}(t),\forall t\ge t_0\}=1$. However, from the proof(...$E\big(\sup_{t_0\le t\le T}|x(t)-\overline{x}(t)|^2\big)=0$) of existence and uniqueness theorem(cf. Stochastic Differential Equations and Applications(second Edition) by Xuerong Mao, page 53), we know that for any two solutions $x(t)$, $\overline{x}(t)$, there is a $P$-null set $\Omega_0(x,\overline{x})$ such that $\forall\omega\in\Omega\setminus\Omega_0(x,\overline{x})$, $x(t)=\overline{x}(t), \forall t\ge t_0$, if there are another two solutions $y(t)$, $\overline{y}(t)$, the $P$-null set $\Omega_0(y,\overline{y})$ may not equal to $\Omega_0(x,\overline{x})$.

It is straightforward to see that \begin{equation}\Omega_0=\bigcup_{x,y\ are\ solutions\ of\ (*)} \Omega_0(x,y)\end{equation}

**My question is:** whether $P(\Omega_0)=0$. In other words, whether we have a unique solution to the following equation for almost surely fixed $\omega$?

\begin{equation}x(t,\omega)=x(t_0,\omega)+\int_{t_0}^tf(x(s,\omega),s)ds+\int_{t_0}^tg(x(s),s)dB(s)(\omega)\end{equation}

Of course, if $P(\Omega_0)\ne0$, we can construct two new processes destroying the pathwise uniqueness, but are these prosesses $\mathscr{F}_t$-adapted?

Thank you.