Hausdorff Dimension - Formal Definition

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).