Finitely-additive set functions naturally arise in mathematical models; however, extending them to \(\sigma \) -additive measures is not always straightforward. For example, a finitely-additive measure defined on rational intervals may require space completion to achieve \(\sigma \) -additivity. A common approach involves the use of Stone spaces, which introduce a totally disconnected topology in order to ensure countable additivity. However, such methods often rely on non-constructive principles. In this paper, we propose a novel and intuitive framework for extending finitely-additive set functions by expanding the space, thereby clarifying the relationship between topological and measure-theoretic extensions.