Measure Theory Of Statistical Convergence

The notion of statstical convergence was introduced by Fast [43] in 1951. From then on, statistical convergence had been investigated and developed in a sequence of articals(see, for instance, [1-20,45-52,48,49,74,75,21-42,53-56,58-64,66,67,72,73,76-98,100-102,104-131]. with the development of statistical convergence, the question of establishing measure theory for statistical convergence has been moving closer to center stage, since a kind of reasonable theory is not only fundamental for unifying various kinds of statistical convergence, but also a bridge linking the study of statistical convergence across measure theory, integration theory, probability and statistics. For this reason, this paper shows many theorems as follows.1 For all finitely additive probability measures defined on theσ-algebra (?) of all subsets of N.2 Proves that every such measure can be uniquely decomposed into a convex combination of a countably additive probability measure and a statistical measure (i.e. a finitely additive probability measureμwithμ(k) = 0 for all singletons {k} ).This paper also shows that classical statistical measures have many nice properties, such as:3 The set (?) of all such measures endowed with the topology of point-wise convergence on (?) forms a compact convex Hausdorff space.4 Every classical statistical measure is of continuity type (hence, atom-less).5 Every specific class of statistical measures fits a complementation minimax rule for every subset in N.6 This paper shows that every kind of statistical convergence can be unified in convergence of statistical measures.