Lebesgue's expansion for spherically invariant measures

Let $H$ be a real complete separable Hilbert space, $\mathscr H$ be the Borel $\sigma$-algebra on $H$, $\mathscr P_S$ be a family of probability measureson $\{H,\mathscr H\}$. Let the characteristic functional of every measure belonging to $\mathscr P_S$ may be represented in the form
$$ \chi(v)=\int_0^\infty\exp\Bigl\{j(b,v)-\frac x2(Kv,v)\Bigr\}\nu(dx), $$
where $b\in H$, $(Kv,v)>0$ for every $v\in H$, $\nu$ is a probability measure on $(0,\infty)$, $\displaystyle\int_0^\infty x\nu(dx)<\infty$. In the paper the Lebesgue's expansion for the pair of measures $\mathbf P_1$, $\mathbf P\in\mathscr P_S$ is derived.