On the extension and kernels of signed bimeasures and their role in stochastic integration

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).