Given a real number we are going to define a Borel external measure on with values in which will comprehend and generalize the concepts of length (for ), area () and volume () of sets in . In particular if is an -dimensional regular surface then one will show that is the -dimensional area of . However, being an external measure, is defined not only on regular surfaces but on every subset of thus generalizing the concepts of length, area and volume. In particular, for , it turns out that the Hausdorff measure is nothing else than the Lebesgue measure of .
Given any fixed set one can consider the measures with varying in . We will see that for a fixed set there exists at most one value such that is finite and positive; while for every other value one will have if and if . For example, if is a regular -dimensional surface then only (which is the area of the surface) may possibly be finite and different from while, for example, the volume of will be and the length of will be infinite.
This can be used to define the dimension of a set (this is called the Hausdorff dimension). A very interesting fact is the existence of sets with dimension which is not integer, as happens for most fractals.
Also, the measure is naturally defined on every metric space , not only on .
Let be a metric space. Given we define the diameter of as
Given a real number we consider the conventional constant
where is the gamma function.
For all , and let us define
The infimum is taken over all possible enumerable families of sets which are sufficiently small () and which cover .
Notice that the function is decreasing in . In fact given the family of sequences considered in the definition of contains the family of sequences considered in the definition of and hence the infimum is smaller. So the limit in the following definition exists:
The number is called -dimensional Hausdorff measure of the set .
|Date of creation||2013-03-22 14:27:26|
|Last modified on||2013-03-22 14:27:26|
|Last modified by||paolini (1187)|