# Hausdorff measure

## Introduction

Given a real number $\alpha \ge 0$ we are going to define a Borel external measure^{} ${\mathscr{H}}^{\alpha}$ on ${\mathbb{R}}^{n}$ with values in $[0,+\mathrm{\infty}]$ which will comprehend and generalize the concepts of length (for $\alpha =1$), area ($\alpha =2$) and volume ($\alpha =3$) of sets in ${\mathbb{R}}^{n}$.
In particular if $M\subset {\mathbb{R}}^{n}$ is an $m$-dimensional regular surface then one will show that
${\mathscr{H}}^{m}(M)$ is the $m$-dimensional area of $M$.
However, being an external measure, ${\mathscr{H}}^{m}$ is defined not only on regular surfaces but on every subset of ${\mathbb{R}}^{n}$ thus generalizing the concepts of length, area and volume. In particular, for $m=n$, it turns out that the Hausdorff measure^{} ${\mathscr{H}}^{n}$ is nothing else than the Lebesgue measure^{} of ${\mathbb{R}}^{n}$.

Given any fixed set $E\subset {\mathbb{R}}^{n}$ one can consider the measures ${\mathscr{H}}^{\alpha}(E)$ with $\alpha $ varying in $[0,+\mathrm{\infty})$. We will see that for a fixed set $E$ there exists at most one value $\alpha $ such that ${\mathscr{H}}^{\alpha}(E)$ is finite and positive; while for every other value $\beta $ one will have ${\mathscr{H}}^{\beta}(E)=0$ if $\beta >\alpha $ and ${\mathscr{H}}^{\beta}(E)=+\mathrm{\infty}$ if $$. For example, if $E$ is a regular $2$-dimensional surface then only ${\mathscr{H}}^{2}(E)$ (which is the area of the surface) may possibly be finite and different from $0$ while, for example, the volume of $E$ will be $0$ and the length of $E$ will be infinite.

This can be used to define the dimension of a set $E$ (this is called the Hausdorff dimension^{}). A very interesting fact is the existence of sets with dimension $\alpha $ which is not integer, as happens for most *fractals ^{}*.

Also, the measure ${\mathscr{H}}^{\alpha}$ is naturally defined on every metric space $(X,d)$, not only on ${\mathbb{R}}^{n}$.

## Definition

Let $(X,d)$ be a metric space. Given $E\subset X$ we define the diameter of $E$ as

$$\mathrm{diam}(E):=\underset{x,y\in E}{sup}d(x,y).$$ |

Given a real number $\alpha $ we consider the conventional constant

$${\omega}_{\alpha}=\frac{{\pi}^{\alpha /2}}{\mathrm{\Gamma}(\alpha /2+1)}$$ |

where $\mathrm{\Gamma}(x)$ is the gamma function^{}.

For all $\delta >0$, $\alpha \ge 0$ and $E\subset X$ let us define

$${\mathscr{H}}_{\delta}^{\alpha}(E):=inf\{\sum _{j=0}^{\mathrm{\infty}}{\omega}_{\alpha}{\left(\frac{\mathrm{diam}({B}_{j})}{2}\right)}^{\alpha}:{B}_{j}\subset X,\bigcup _{j=0}^{\mathrm{\infty}}{B}_{j}\supset E,\mathrm{diam}({B}_{j})\le \delta \forall j=0,1,\mathrm{\dots}\}.$$ | (1) |

The *infimum* is taken over all possible enumerable families of sets ${B}_{0},{B}_{1},\mathrm{\dots},{B}_{j},\mathrm{\dots}$ which are sufficiently small ($\mathrm{diam}{B}_{j}\le \delta $) and which cover $E$.

Notice that the function^{} ${\mathscr{H}}_{\delta}^{\alpha}(E)$ is decreasing in $\delta $. In fact given ${\delta}^{\prime}>\delta $ the family of sequences ${B}_{j}$ considered in the definition of ${\mathscr{H}}_{{\delta}^{\prime}}^{\alpha}$ contains the family of sequences considered in the definition of ${\mathscr{H}}_{\delta}^{\alpha}$ and hence the infimum is smaller.
So the limit in the following definition exists:

$${\mathscr{H}}^{\alpha}(E):=\underset{\delta \to {0}^{+}}{lim}{\mathscr{H}}_{\delta}^{\alpha}(E).$$ | (2) |

The number ${\mathscr{H}}^{\alpha}(E)\in [0,+\mathrm{\infty}]$ is called *$\alpha $-dimensional Hausdorff measure* of the set $E\subset X$.

Title | Hausdorff measure |
---|---|

Canonical name | HausdorffMeasure |

Date of creation | 2013-03-22 14:27:26 |

Last modified on | 2013-03-22 14:27:26 |

Owner | paolini (1187) |

Last modified by | paolini (1187) |

Numerical id | 8 |

Author | paolini (1187) |

Entry type | Definition |

Classification | msc 28A78 |

Related topic | HausdorffDimension |

Related topic | LebesgueMeasure |