Extension of measures and stochastic equations

M. P. Ershov

Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 3, pp. 457-471

Résumé

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.