In
mathematics, Hausdorff measure is a generalization of the traditional notions of
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open s ...
and
volume to non-integer dimensions, specifically
fractals and their
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of ...
s. It is a type of
outer measure, named for
Felix Hausdorff
Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, and f ...
, that assigns a number in
,∞to each set in
or, more generally, in any
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
.
The zero-dimensional Hausdorff measure is the number of points in the set (if the set is finite) or ∞ if the set is infinite. Likewise, the one-dimensional Hausdorff measure of a
simple curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
in
is equal to the length of the curve, and the two-dimensional Hausdorff measure of a
Lebesgue-measurable subset of
is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes the
Lebesgue measure and its notions of counting, length, and area. It also generalizes volume. In fact, there are ''d''-dimensional Hausdorff measures for any ''d'' ≥ 0, which is not necessarily an integer. These measures are fundamental in
geometric measure theory. They appear naturally in
harmonic analysis or
potential theory.
Definition
Let
be a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
. For any subset
, let
denote its diameter, that is
:
Let
be any subset of
and
a real number. Define
:
where the infimum is over all countable covers of
by sets
satisfying
.
Note that
is monotone nonincreasing in
since the larger
is, the more collections of sets are permitted, making the infimum not larger. Thus,
exists but may be infinite. Let
:
It can be seen that
is an
outer measure (more precisely, it is a
metric outer measure). By
Carathéodory's extension theorem
In measure theory, Carathéodory's extension theorem (named after the mathematician Constantin Carathéodory) states that any pre-measure defined on a given ring of subsets ''R'' of a given set ''Ω'' can be extended to a measure on the σ-al ...
, its restriction to the σ-field of
Carathéodory-measurable sets is a measure. It is called the
-dimensional Hausdorff measure of
. Due to the
metric outer measure property, all
Borel subsets of
are
measurable.
In the above definition the sets in the covering are arbitrary. However, we can require the covering sets to be open or closed, or in
normed spaces
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...
even convex, that will yield the same
numbers, hence the same measure. In
restricting the covering sets to be balls may change the measures but does not change the dimension of the measured sets.
Properties of Hausdorff measures
Note that if ''d'' is a positive integer, the ''d''-dimensional Hausdorff measure of
is a rescaling of the usual ''d''-dimensional
Lebesgue measure , which is normalized so that the Lebesgue measure of the unit cube
,1sup>''d'' is 1. In fact, for any Borel set ''E'',
:
where α
''d'' is the volume of the unit
''d''-ball; it can be expressed using
Euler's gamma function
:
This is
:
,
where
is the volume of the unit diameter ''d''-ball.
Remark. Some authors adopt a definition of Hausdorff measure slightly different from the one chosen here, the difference being that the value
defined above is multiplied by the factor
, so that Hausdorff ''d''-dimensional measure coincides exactly with Lebesgue measure in the case of Euclidean space.
Relation with Hausdorff dimension
It turns out that
may have a finite, nonzero value for at most one
. That is, the Hausdorff Measure is zero for any value above a certain dimension and infinity below a certain dimension, analogous to the idea that the area of a line is zero and the length of a 2D shape is in some sense infinity. This leads to one of several possible equivalent definitions of the Hausdorff dimension:
:
where we take
and
.
Note that it is not guaranteed that the Hausdorff measure must be finite and nonzero for some ''d'', and indeed the measure at the Hausdorff dimension may still be zero; in this case, the Hausdorff dimension still acts as a change point between measures of zero and infinity.
Generalizations
In
geometric measure theory and related fields, the
Minkowski content The Minkowski content (named after Hermann Minkowski), or the boundary measure, of a set is a basic concept that uses concepts from geometry and measure theory to generalize the notions of length of a smooth curve in the plane, and area of a smoot ...
is often used to measure the size of a subset of a metric measure space. For suitable domains in Euclidean space, the two notions of size coincide, up to overall normalizations depending on conventions. More precisely, a subset of
is said to be
-rectifiable if it is the image of a
bounded set
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis and related areas of m ...
in
under a
Lipschitz function. If