In mathematics, the box-counting content is an analog of Minkowski content.
Definition
Let be a bounded subset of -dimensional Euclidean space such that the box-counting dimension exists. The upper and lower box-counting contents of are defined by
where is the maximum number of disjoint closed balls with centers and radii .
If , then the common value, denoted , is called the box-counting content of .
If , then is said to be box-counting measurable.
Examples Let denote the unit interval. Note that the box-counting dimension and the Minkowski dimension coincide with a common value of 1; i.e.
Now observe that , where denotes the integer part of . Hence is box-counting measurable with .
By contrast, is Minkowski measurable with .
See also Box counting
References