On a characterization of certain families of measures

Let $(\xi_t)$, $t\in[0,1]$, be a measurable stochastic process. Put
$$ \mu_t(A)=\int_0^t 1_A(\xi_s)\,ds\qquad A\in\mathscr B, $$
where $\mathscr B$ is the Borel $\sigma$-algebra in $R^1$. It is easy to see that the family has the following two properties:
1) for any $A\in\mathscr B$ the function $t\rightsquigarrow\mu_t(A)$ is non-decreasing;
2) for any $t\in[0,1]$, $\mu_t(R^1)=t$.
We give a characterization of families $(\mu_t)$, with properties 1) and 2), which are generated by a stochastic process.