We consider a little-known abstract decomposition result for positive measures due to Dellacherie, and show that it yields many decompositions of measures, several of which are new. We then extend Dellacherie's result to (controlled) vector measures, and apply it to obtain a decomposition of semimartingales due to Bichteler, on which we improve. Then, we investigate how the outputs of the decomposition depend on its inputs, in particular characterizing the two elements of the decomposition as projections in the sense of Riesz spaces and of metric spaces. Finally, we prove a decomposition theorem for strictly positive operators on Riesz spaces which generalizes Dellacherie's Theorem.