  The concept of finitely additive supermartingales, originally due to Bochner,
is revived and developed. We exploit it to study measure decompositions over
filtered probability spaces and the properties of the associated
Dol\'{e}ans-Dade measure. We obtain versions of the Doob Meyer decomposition
and, as an application, we establish a version of the Bichteler and Dellacherie
theorem with no exogenous probability measure.
