A Representation Theorem For Real (Complex)-Valued Measures Of Statistical Type And Its Application To Statistical Convergence

The notion of statstical convergence was introduced by Fast in 1951. From then on, statistical convergence had been investigated and developed in a sequence of articals. 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.A real (complex)-valued finitely additive measureμon N is said to be a measure of statistical type providedμ(k) = 0 for all singletons {k}. This paper shows that the space of all real (complex)-valued finitely additive finite measureμon N endow with the norm of total variation is isometric to the dual of l~∞;and the space of all measures of statistical type is isometric to the dual of l~∞/c_o. As its application, this paper proves that every kind of statistical convergence is just a type of measure convergence with respect to a specific class of statistical measures.