Density of a set
that is measurable on the real line , at a point
The limit (if it exists) of the ratio
where is any segment containing and is its length. If one considers an outer measure instead of a measure, one obtains the definition of the outer density of at . Similarly one can introduce the density in -dimensional space. Here the lengths of the segments in are replaced by the volumes of the corresponding -dimensional parallelepipeds with faces parallel to the coordinate planes, while the limit is considered as the diameters of the parallelepipeds tend to zero. For sets from it is useful to employ the concept of the right (left) density of a set at a point , which is obtained from the general definition if in it one considers only segments having left (right) ends at . Very often, the concept of density is used when the density of the set is equal to one (see Density point) or zero (see Thinness of a set).
|||I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian)|
|||S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)|
See [a1] for a nice topological application of these notions.
|[a1]||F.D. Tall, "The density topology" Pacific J. Math , 62 (1976) pp. 275–284|
Density of a set. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Density_of_a_set&oldid=18377