On real-valued measures of statistical type and their applications to statistical convergence

A real-valued finitely additive measure μμ on NN is said to be a measure of statistical type provided μ(k)=0μ(k)=0 for all singletons {k}{k}. Applying the classical representation theorem of finitely additive measures with totally bounded variation, we first present a short proof of the representation theorem of statistical measures. As its application, we show that every kind of statistical convergence is just a type of measure convergence with respect to a specific class of statistical measures.