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In the paper, the martingales and super-martingales relative to a regular setof measures are systematically studied. The notion of local regularsuper-martingale relative to a set of equivalent measures is introduced and thenecessary and sufficient conditions of the local regularity of it in thediscrete case are founded. The regular set of measures play fundamental rolefor the description of incomplete markets. In the partial case, the descriptionof the regular set of measures is presented. The notion of completeness of theregular set of measures have the important significance for the simplificationof the proof of the optional decomposition for super-martingales. Using thisnotion, the important inequalities for some random values are obtained. Theseinequalities give the simple proof of the optional decomposition of themajorized super-martingales. The description of all local regularsuper-martingales relative to the regular set of measures is presented. It isproved that every majorized super-martingale relative to the complete set ofmeasures is a local regular one. In the case, as evolution of a risk asset isgiven by the discrete geometric Brownian motion, the financial market isincomplete and a new formula for the fair price of super-hedge is founded.

Description of Incomplete Financial Markets for the Discrete Time Evolution of Risk Assets