On statistical measure theory

The purpose of this paper is to unify various kinds of statistical convergence by statistical measure convergence and to present Jordan decomposition of finitely additive measures. It is done through dealing with the most generalized statistical convergence–ideal convergence by applying geometric functional analysis and Banach space theory. We first show that for each type of ideal I(⊂2N)I(⊂2N) convergence, there exists a set SS of statistical measures such that the measure SS-convergence is equivalent to the statistical convergence. To search for Jordan decomposition of measures of statistical type, we show that the subspace XI≡span¯{χA:A∈I} is an ideal of the space ℓ∞ℓ∞ in the sense of Banach lattice, hence the quotient space ℓ∞/XIℓ∞/XI is isometric to a C(K)C(K) space. We then prove that a statistical measure has a Jordan decomposition if and only if its corresponding functional is norm-attaining on ℓ∞ℓ∞, and which in turn induces an approximate null–ideal preserved Jordan decomposition theorem of finitely additive measures. Finally, we show this characterization and the approximate decomposition theorem are true for finitely additive measures defined on a general measurable space.