On statistical measure theory

Statistical convergence and measure convergence generated by a single statistical measure

“…A sequence (x n ) in a topological space X is said to be statistically convergent to x ∈ X provided for any neighborhood U of x, we have χ A(ε) (j) = 0 ∞ (I) ⊂ ∞ and the quotient space ∞ / ∞ (I) for an ideal I. As a result, we show that ∞ (I) is an also algebraic ideal (it is shown in [2] that ∞ (I) is a lattice ideal of ∞ ); and ∞ / ∞ (I) is either finite dimensional, or, containing ∞ . ∞ / ∞ (I) is also characterized by properties of statistical measures corresponding to the ideal I, or, filter F. In the forth section, we present some characterizations of those kinds of filter convergence which are equivalent to statistical measure convergence defined by a single statistical measure.…”

“…The following property can be found in [2]. Proposition 2.1 For any bounded finitely additive measure μ : 2 N → R,…”

“…The measure μ-convergence is equivalent to the ideal I-convergence, which is, according to [2,Theorem 2.12], further equivalent to measure S U -convergence. By the Krein-Milman theorem [34], S U is the w * -closed convex hull of E(S U ).…”

“…It is shown in [2, Lemma 3.1], that for any ideal I ⊂ 2 N , the subspace ∞ (I) (denoted by X I in [2]) is a lattice ideal of ∞ . The following theorem says that ∞ (I) is also an algebraic ideal.…”

“…Therefore, for each x * ∈ ∞ (I) ⊥ , the statistical type measure μ ≡ μ x * defined by (2.4) satisfies μ(A) = 0 for all A ∈ I. According to [2,Theorem 3.4], for every n ∈ N, there are positive measures μ + n , μ − n such that…”