Stochastic integration on the predictable $\sigma$-field with respect to $\sigma$-finite $L^0$-valued measures, also known as formal semimartingales, is studied. In particular, the triplet of such measures is introduced and used to characterize the set of integrable processes. Special attention is given to Lévy processes indexed by the real line. Surprisingly, many of the basic properties break down in this situation compared to the usual $\mathbb{R}_+$ case.

Keywords: stochastic integration; semimartingales; Lévy processes; vector measures

AMS Subject Classification: 60G44; 60G57; 60H05.