𝑅-type summability methods, Cauchy criteria, 𝑃-sets and statistical convergence

A summability method S S is called an R R -type summability method if S S is regular and x y xy is strongly S S -summable to 0 whenever x x is strongly S S -summable to 0 and y y is a bounded sequence. Associated with each R R -type summability method S S are the following two methods: convergence in μ \mu -density and μ \mu -statistical convergence where μ \mu is a measure generated by S S . In this note we extend the notion of statistically Cauchy to μ \mu -Cauchy and show that a sequence is μ \mu -Cauchy if and only if it is μ \mu -statistically convergent. Let W ( A ) = A ¯ β N ∩ β N ∖ N W\left ( A \right ) = {\overline A ^{\beta \mathbb {N}}} \cap \beta \mathbb {N}\backslash \mathbb {N} for A ⊂ N A \subset \mathbb {N} and K = ∩ { W ( A ) : A ⊆ N , χ A is strongly S − summable to 1 } \mathcal {K}{\text { = }} \cap \left \{ {W\left ( A \right ):A \subseteq \mathbb {N}{\text {,}}{\chi _A}\;{\text {is}}\;{\text {strongly}}\;S - {\text {summable}}\;{\text {to}}\;1} \right \} . Then μ \mu -Cauchy is equivalent to convergence in μ \mu -density if and only if every G δ {G_\delta } that contains K \mathcal {K} in β N ∖ N \beta \mathbb {N}\backslash \mathbb {N} is a neighborhood of K \mathcal {K} in β N ∖ N \beta \mathbb {N}\backslash \mathbb {N} . As an application, we show that the bounded strong summability field of a nonnegative regular matrix admits a Cauchy criterion.