# Definition:Hausdorff-Besicovitch Dimension

## Definition

Let $\R^n$ be the $n$-dimensional Euclidean space.

Let $F \subseteq \R^n$.

The Hausdorff-Besicovitch dimension of $F$ is defined as:

 $\ds \map {\dim_H} F$ $:=$ $\ds \inf \set {s \in \R_{\ge 0} : \map {\HH^s} F = 0}$ $\ds$ $=$ $\ds \sup \set {s \in \R_{\ge 0} : \map {\HH^s} F = +\infty}$

where $\map {\HH^s} \cdot$ denotes the $s$-dimensional Hausdorff measure on $\R^n$.

## Also known as

This is also known as the Hausdorff dimension.

## Source of Name

This entry was named for Felix Hausdorff and Abram Samoilovitch Besicovitch.