  In the paper, the martingales and super-martingales relative to a regular set
of measures are systematically studied. The notion of local regular
super-martingale relative to a set of equivalent measures is introduced and the
necessary and sufficient conditions of the local regularity of it in the
discrete case are founded. The regular set of measures play fundamental role
for the description of incomplete markets. In the partial case, the description
of the regular set of measures is presented. The notion of completeness of the
regular set of measures have the important significance for the simplification
of the proof of the optional decomposition for super-martingales. Using this
notion, the important inequalities for some random values are obtained. These
inequalities give the simple proof of the optional decomposition of the
majorized super-martingales. The description of all local regular
super-martingales relative to the regular set of measures is presented. It is
proved that every majorized super-martingale relative to the complete set of
measures is a local regular one. In the case, as evolution of a risk asset is
given by the discrete geometric Brownian motion, the financial market is
incomplete and a new formula for the fair price of super-hedge is founded.
