Abstract
In keeping with our general approach to the concept of a stochastic process we will put definitions and first facts pertaining to J. L. Doob's semi-martingales into rather general settings, and the purpose in so doing is to clarify and simplify rather than to generalize. Throughout we will be dealing with algebras of sets, probability measures and other set functions either of which will be finitely additive or countably additive, depending on the context. We are emphasizing that as a rule they are meant to be finitely additive only, unless countable additivity is expressly stipulated, or is unavoidable in the context.
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