Decomposition theorem and its applications

In the article a complete proof of decomposition theorem is given. This theorem concerns the so called canonical extension of the order relation on the set of probabilistic measures. Here we study a structure for an extension of the order relation given on some set $A$ on the generated vector space $\mathbb{R}^A$. Corresponding description of the extension with help of stochastic matrices is found (Theorem 2). Decomposition theorem reveals the most significant properties of the canonical extension of orders. In particular the consequences of the theorem are two important statements:

• The coincidence of the canonical extension of any order with its convex hull and
• Truth of Choquet condition for the canonical extension (see the corollary 1 and the corollary 2 in the section 3.1).

The complete proof of decomposition theorem is quite complicated. As the first step for the proof of this theorem we prove an assertion of existence of optimal surplus vector (Theorem 1). This theorem having game-theoretical interpretation also can be formulated in economic terms (Remark 1). A geometric interpretation of decomposition theorem is given (example 1).