On existence and a dominated convergence theorem for weighted $g$-summability

Abstract:

Let $({w_1},{w_2},{w_3})$ be an ordered triple of real numbers such that ${w_1} + {w_2} + {w_3} = 1$. Let g be a real-valued function on the entire real axis which is of bounded variation on every closed interval. For f a real-valued function on the entire real axis which is bounded on a closed interval [a, b], we use the F. Riesz step function approach to define the concept of f being $({w_1},{w_2},{w_3})$ g-summable over [a, b], and we define the integral \[ [F,({w_1},{w_2},{w_3})]s\int _a^b {f(x)dg(x)} \] when f has this property. We show that this integral extends the weighted refinement integral $[F,({w_1},{w_2},{w_3})]\int _a^b {f(x)dg(x)}$ for f’s as above. This paper generalizes the method of Pasquale Porcelli for the Stieltjes mean sigma integral. We present an existence theorem for the integral defined here involving saltus and continuous parts of g. We establish a convergence theorem for this integral which is analogous to the Lebesgue Dominated Convergence Theorem for the Lebesgue-Stieltjes integral.