Theorem: If $\{X(t)\}_{t \in [0,\infty)}$ is a continuous martingale of finite variation then almost surely it holds that $X(t)=X(0)$ for all $t \in [0,\infty)$.
Does this mean that almost every path is constant with the same constant, i.e. $X (t,\omega_1)=const.= X(t,\omega_2)$? Or does it mean that almost every path is constant but the constants of two paths can be different ones?
The second one. Consider $X(0)$ being any non-trivial random variable and letting $X(t) = X(0)$ for all $t$. It is clear this is a continuous martingale of finite variation, but the constant for each path is different.