  In this work we provide a necessary and sufficient condition for the
extension of signed bimeasures on $\delta$-rings and for the existence of
relative kernels. This result generalises the construction method of regular
conditional probabilities to the more general setting of extended signed
measures. Building on this result, we obtain the most general theory of
stochastic integrals based on random measures, thus extending and generalising
the whole integration theory developed in the celebrated Rajput and Rosinski's
paper (\textit{Probab.~Theory Relat.~Fields}, \textbf{82} (1989) 451-487) and
the recent results by Passeggeri (\textit{Stoch.~Process.~Their Appl.},
\textbf{130}, (3), (2020), 1735-1791).
