mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the notion of an (exact) dimension function (also known as a gauge function) is a tool in the study of

fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...

s and other subsets of

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 settin ...

s. Dimension functions are a generalisation of the simple "

diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...

to the

dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...

power law In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a proportional relative change in the other quantity, inde ...

used in the construction of ''s''-dimensional

Hausdorff measure In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that ass ...

Motivation: ''s''-dimensional Hausdorff measure

Consider a metric space (''X'', ''d'') and a

subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

''E'' of ''X''. Given a number ''s'' ≥ 0, the ''s''-dimensional Hausdorff measure of ''E'', denoted ''μ''^''s''(''E''), is defined by :

\mu^ (E) = \lim_ \mu_^ (E),

where :

\mu_^ (E) = \inf \left\.

''μ''_''δ''^''s''(''E'') can be thought of as an approximation to the "true" ''s''-dimensional area/volume of ''E'' given by calculating the minimal ''s''-dimensional area/volume of a covering of ''E'' by sets of diameter at most ''δ''. As a function of increasing ''s'', ''μ''^''s''(''E'') is non-increasing. In fact, for all values of ''s'', except possibly one, ''H''^''s''(''E'') is either 0 or +∞; this exceptional value is called the Hausdorff dimension of ''E'', here denoted dim_H(''E''). Intuitively speaking, ''μ''^''s''(''E'') = +∞ for ''s'' < dim_H(''E'') for the same reason as the 1-dimensional linear

length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...

of a 2-dimensional disc in the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...

is +∞; likewise, ''μ''^''s''(''E'') = 0 for ''s'' > dim_H(''E'') for the same reason as the 3-dimensional

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...

of a disc in the Euclidean plane is zero. The idea of a dimension function is to use different functions of diameter than just diam(''C'')^''s'' for some ''s'', and to look for the same property of the Hausdorff measure being finite and non-zero.

Definition

Let (''X'', ''d'') be a metric space and ''E'' ⊆ ''X''. Let ''h'' : , +∞) → [0, +∞be a function. Define ''μ''^''h''(''E'') by :

\mu^ (E) = \lim_ \mu_^ (E),

where :

\mu_^ (E) = \inf \left\.

Then ''h'' is called an (exact) dimension function (or gauge function) for ''E'' if ''μ''^''h''(''E'') is finite and strictly positive. There are many conventions as to the properties that ''h'' should have: Rogers (1998), for example, requires that ''h'' should be

monotonically increasing In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...

for ''t'' ≥ 0, strictly positive for ''t'' > 0, and continuous function">continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...

Motivation: ''s''-dimensional Hausdorff measure

Definition

Packing dimension

Example

References