AbstractA certain type of partial orderings of positive measures has recently attracted attention in connection with Choquet's theorem. It had been known in special cases for a long time that these orderings have the property: If a measure v is “more diffuse” than a sum of measures ∑ μi, then v can be written as a sum v = ∑ vi such that each summand vi is more diffuse than the corresponding μi. A similar problem of decomposition can be formulated for instance for convex functions on some convex set: Let l be convex with l ⩾ ∑ ki where the ki are convex. Do there exist convex li with ∑ li = l and ki ⩽ li ?The present paper gives a systematic account on problems of this type. It develops a fairly general method of finding inequalities for the given quantities (v, μ1,…, μn, respectively, l, k1,…, kn) which are necessary and sufficient for decomposability.A powerful generalization of the Hahn-Banach theorem plays the crucial part in the proof of existence. It seems to be of importance in other contexts too and is discussed in some detail.Measure theoretic extensions of the existence arguments suggested in particular by V. Strassen's work are not treated in the present paper. However, some results along these lines obtained before by ad hoc methods are listed in the introduction, to give an idea of the relevance of decomposition problems.The paper is completely self-contained.