Extension of measures and stochastic equations

E. Szpilrajn-Marczewski [2] constucted a measure on a sub-$\sigma$-algebra of the Borel $\sigma$-algebra in a complete separable metric space which could not be extended to the Borel $\sigma$-algebra. That measure was not separable. In connection with this example, E. Szpilrajn-Marczewski [3] posed the following problem: whether any separable measure on the $\sigma$-algebra generated by a family of Borel sets in a complete separable metric sprace can be extended to the whole Borel $\sigma$-algebra.
In the paper, this problem is answered, in general, negatively. However, it is proved that an extension does exist under the condition that the $\sigma$-algebra the original measure is defined on is countably generated.
The problem of extending a measure is shown to be equivalent to that of solving a stochastic equation: given a measurable mapping $F$ of a measurable space ($X$, $\mathscr X$) into a measure space ($Y$, $\mathscr Y$, $\nu$), a measure $\mu$ on ($X$, $\mathscr X$) is called a solution of the stochastic equation
$$ F\circ\mu=\nu $$
if, for any $B\in\mathscr Y$ $\mu(F^{-1}\circ B)=\nu B$.
For sufficiently “respectable” spaces ($X$, $\mathscr X$) and ($Y$, $\mathscr Y$), the condition
$$ (F^{-1}\circ B)\ne\varnothing\quad\forall B\in\mathscr Y\colon\nu(B)>0 $$
is proved to be sufficient (and obviously necessary) for the equation $F\circ\mu=\nu$ to have at least one solution.
The problem of uniqueness of a solution of the equation $F\circ\mu=\nu $, respectively, of an extension of a given measure is also investigated.