Formal Definition
Let be a metric space. If and, the -dimensional Hausdorff content of is defined by
In other words, is the infimum of the set of numbers such that there is some (indexed) collection of balls covering with for each which satisfies . (Here, we use the standard convention that inf Ø =∞.) The Hausdorff dimension of is defined by
Equivalently, may be defined as the infimum of the set of such that the -dimensional Hausdorff measure of is zero. This is the same as the supremum of the set of such that the -dimensional Hausdorff measure of is infinite (except that when this latter set of numbers is empty the Hausdorff dimension is zero).
Read more about this topic: Hausdorff Dimension
Famous quotes containing the words formal and/or definition:
“The bed is now as public as the dinner table and governed by the same rules of formal confrontation.”
—Angela Carter (1940–1992)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)