Remarks on factorization property of some stochastic integrals

In the paper Sato (2006) there are introduced two families of improper random integrals and the corresponding two convolution semigroups of infinitely divisible laws on ${\text{Rset}}^d$. Theorem 3.1 gives a relation (a factorization property) between those two integrals. Here, using \emph{the random integral mappings} $I_{(a,b]}^{h,r}$ (cf. the survey article Jurek (2011)), we give a simpler proof that is also valid for measures on Banach spaces. Furthermore, using our technique we establish yet other relations between those two families of improper stochastic integrals.