Techniques to Calculate Hausdorff Measure

Mass Distribution

A measure on a bounded subset of for which will be called a mass distribution.

Mass Distribution Principle

Let be a mass distribution on , and suppose for some , there are numbers and such that for all sets with . Then and

Proof Suppose is a cover for , , then Taking infimum, we have for sufficiently small . Thus, . In particular, since , we get .

Potential Theoretic Methods

-Potential

For some , the -potential at a point , resulting from the mass distribution on is defined as

This is a Generalization of The Gravitational Potential

If we are working in and , then this is essentially the familiar Newtonian gravitational potential.

-Energy

The -energy of a mass distribution on is defined as

Theorem

Let be a subset of .

• If there is a mass distribution on with , then and .
• If is a Borel set with , then for all , there exists a mass distribution on with .

Proof