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A measure theoretical subsequence characterization of statistical convergence. (English) Zbl 0830.40002
Let $$T= (a_{mn})$$ be a triangular matrix with $$a_{mn}>0$$ for $$n\leq m$$, $$\sum_n a_{mn}= 1$$ $$(m\in \mathbb{N})$$ and $$\lim_m a_{mn}= 0$$ $$(n\in \mathbb{N})$$. A sequence of reals $$S= \{s_n\}$$ is said to be statistically $$T$$-summable to $$L$$ (briefly $$s_n\to L$$ (stat $$T$$)) if $$\sum_n [a_{mn}$$: $$|s_n- L|\geq \varepsilon ]\to 0$$ $$(m\to \infty)$$ for every $$\varepsilon>0$$. A subset $$A\subset \mathbb{N}$$ is said to have $$T$$-density zero if $$\lim_m \sum_{n\in A} a_{mn} =0$$.
For a sequence $$S$$ let $$S(x)$$ denote the subsequence of $$S$$ with $$s_n\in S(x) \iff e_n (x) =1$$ where $$\sum_n e_n (x) 2^{-n}$$ is the binary expansion of $$x\in (0, 1]$$ with $$e_n (x)\in \{0, 1\}$$.
The main result of the paper is that $$s_n\to L$$ (stat $$T$$) if and only if there exists a subset $$A= \{k_n\}$$ of $$\mathbb{N}$$ having $$T$$-density zero such that $$m_A (\{x\in (0, 1]$$: $$\lim_n (S(x ))_n= L\})=1$$. Here $$m_A$$ is the unique probability measure defined on the Borel subsets of $$(0,1]$$ having the following property: $m_A (\{x\in (0,1]:\;e_j (x)= 1\})= \begin{cases} {1\over 2} &\text{if } j\not\in A\\ {1\over 2^n} &\text{if } j= k_n \end{cases} ,$ and $$\{e_j (x)\}$$ is a sequence of independent random variables with respect to $$m_A$$.
Reviewer: T.Leiger (Tartu)

MSC:
 40A05 Convergence and divergence of series and sequences 40D25 Inclusion and equivalence theorems in summability theory 28A12 Contents, measures, outer measures, capacities 40G99 Special methods of summability
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