Boundedness of Sequentially Generated Measurable Products: A Neighborhood Approach

Many studies have been done on products of measurable sets. The most recent results highlight the properties of tensor products expressed as matrix products. Therefore, this study builds on existing research on product of measurable sets focusing on properties expressed in matrix products. This study investigates the conditions under which sequentially generated products of functions are measurably bound using (∈ - δ) criterion for uniform continuity . This article explores the connection between topological properties of measurable sets and boundedness of their products. The study sheds light on the application of r-neighborhood topological properties of refinement of measurable sets in determining the boundedness of sequentially generated products of measurable functions. Concepts such as monotonicity of functions, continuity from above of set functions, almost everywhere properties and r-neighborhood partition of measurable sets are applied in the context of p-integrable functions. The results of this research can be applied to develop the r-neighborhood business models where r represents the physical distance around a fixed business focal point that geographically creates a fruitful business environment for achievement of the optimal industrial and commercial profit margins determined by the boundedness of product functions. For a fixed product of functions i.e. the target of achievement, one can sequentially and by monotonicity of measurable functions determine the quantitative (or measurable) convergence of the product of functions which represents the interactive operational activities towards the defined business goals. Further, the results of this study can be applied in developing geometrical models in engineering by quantitative approximation to desired standards.