Science in China Series A: Mathematics Dec., 2008, Vol. 51, No. 12, 2285–2303 www.scichina.com math.scichina.com www.springerlink.com Measure theory of statistical convergence CHENG LiXin † , LIN GuoChen, LAN YongYi & LIU Hui Department of Mathematics, Xiamen University, Xiamen 361005, China (email: lxcheng@xmu.edu.cn, lanyongyi@jmu.edu.cn, liuhui4452469@gmail.com) 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. Keywords: statistical convergence, statistical measure, subdifferential, Banach space MSC(2000): 60B05, 46G99, 40A99 1 Introduction The notion of statistical convergence was introduced by Fast in [1] in 1951 and was investigated by Connor in [2–10], Fridy, Miller and Orhan in [11–20] and many other mathematicians (see, for instance, [21–122]). Statistical convergence has become an active area of research under the name of statistical convergence since the 1990s of the last century. It has appeared in a wide variety of topics. For example, statistical convergence has been discussed in summability of matrix, series and integral in [3, 4, 5, 13, 32, 36, 61, 94, 110, 123], Fourior analysis in [70, 71, 84], approximation of positive operators in [39–43, 45, 46, 49, 50, 54, 86], number theory in [9], trigonometric series in [38, 96, 122, 124], Banach space theory in [6, 55, 119], locally convex spaces in [24, 91, 92, 125, 126], structure of ideals of bounded continuous functions in [9], fuzzy mathematics in [80, 83, 101], property of continuous functions in [8, 56] and making various new types of topological linear spaces in [58, 61, 102, 113, 115]. For any subset A of the natural numbers set N , let A ♯ denote the cardinal number of A, and A n ={k∈ A : k n} (n=1, 2, ...). Now, we recall the definition of the statistical convergence. A sequence {x k } in a Banach space X is said to be statistically convergent to x ∈ X , if ∀ ε> 0, Received June 21, 2007; accepted September 19, 2007; published online September 13, 2008 DOI: 10.1007/s11425-008-0017-z † Corresponding author This work was supported by the National Natural Science Foundation of China (Grant Nos. 10771175, 10471114)