  In the paper, we introduce the notion of a local regular supermartingale
relative to a convex set of equivalent measures and prove for it the necessary
and sufficient conditions of optional Doob decomposition in the discrete case.
This Theorem is a generalization of the famous Doob decomposition onto the case
of supermartingales relative to a convex set of equivalent measures. The
description of all local regular supermartingales relative to a convex set of
equivalent measures is presented. A notion of complete set of equivalent
measures is introduced. We prove that every non negative bounded
supermartingale relative to a complete set of equivalent measures is local
regular. A new definition of fair price of contingent claim in incomplete
market is given and a formula for fair price of Standard option of European
type is found.
