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

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