Measure theory of statistical convergence

Abstract

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 studies of statistical convergence across measure theory, integration theory, probability and statistics. For this reason, this paper, in terms of subdifferential, first shows a representation theorem for all finitely additive probability measures defined on the σ-algebra A of all subsets of N , and 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 measure μ with μ(k) = 0 for all singletons {k}). This paper also shows that classical statistical measures have many nice properties, such as: The set S of all such measures endowed with the topology of point-wise convergence on A forms a compact convex Hausdorff space; every classical statistical measure is of continuity type (hence, atomless), and every specific class of statistical measures fits a complementation minimax rule for every subset in N . Finally, this paper shows that every kind of statistical convergence can be unified in convergence of statistical measures.

Statistical convergence and measure convergence generated by a single statistical measure

“…It is shown in [10] that every classical statistical measure in ∂ e K • χ is atomless, where · K is defined on ∞ for x = (x(n)) by…”

A functional characterization of measures and the Banach–Ulam problem

“…The corresponding measure μ f is also called a measure of statistical type [2,3] whenever the functional f ∈ c ⊥ 0 .…”

𝐈𝟐 − 𝐒𝐓𝐀𝐓𝐈𝐒𝐓𝐈𝐂𝐀𝐋 And 𝐈𝟐 −Lacunary Statistical Convergence for Double Sequence of Order �

“…Although statistical convergence was introduced over fifty years ago, it has become an active area of research in recent years. This has been applied in various areas such as summability theory (Fridy [13] and Salat [22]), topological groups (Cakalli [2], [3]), topological spaces (Maio and Kocinac [16]), locally convex spaces (Maddox [17]), measure theory (Cheng et al [4]), (Connor and Swardson [7]) and (Miller [19]), Fuzzy Mathematics (Nuray and Savas [21] and Savas [23]). In recent years generalization of statistical convergence has appeared in the study of strong summability and the structure of ideals of bounded functions, (Connor and Swardson [7]).…”