On separation in measure and metric density in Romanovski spaces

In this paper, the notion of metric separation on the line is extended to a notion of separation in measu re in a Romanovski space. It is shown that a set E c X c is a measurable if and only if E and E are separated in measure. Two forms of Lebesque density are discussed, and a Lebesque density theorem is proven for measurable and non-mensurable sets. A study is made, in terms of points of density and dispersion, of when two sets Ej c x and E2 c X are separated in measure and of what happens when they are not. It is shown that almost every point of Lebesque density in one of our senses is a point of Lebesque density in the other sense and conversely. It is shown that the density functions are measurable. Throughout this work, the notation, definitions and con ventions of (7) are adopted.