# Dimension function

In mathematics, the notion of an (exact) dimension function (also known as a gauge function) is a tool in the study of fractals and other subsets of metric spaces. Dimension functions are a generalisation of the simple "diameter to the dimension" power law used in the construction of s-dimensional Hausdorff measure.

## Motivation: s-dimensional Hausdorff measure

Consider a metric space (X, d) and a subset E of X. Given a number s ≥ 0, the s-dimensional Hausdorff measure of E, denoted μ$^s$(E), is defined by
    $\mu ^{s}(E)=\lim _{\delta \to 0}\mu _{\delta }^{s}(E),$
where
    $\mu _{\delta }^{s}(E)=\inf \left\{\left.\sum _{i=1}^{\infty }\mathrm {diam} (C_{i})^{s}\right|\mathrm {diam} (C_{i})\leq \delta ,\bigcup _{i=1}^{\infty }C_{i}\supseteq E\right\}.$
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$(E) for the same reason as the 1-dimensional linear length of a 2-dimensional disc in the Euclidean plane is +∞; likewise, μ$^s$(E) = 0 for s > dim$_H$(E) for the same reason as the 3-dimensional volume of a disc in the Euclidean plane is zero.
The idea of a dimension function is to use different functions of diameter than just diam(C)$^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, +∞) → [0, +∞] be a function. Define μ$^h$(E) by
    $\mu ^{h}(E)=\lim _{\delta \to 0}\mu _{\delta }^{h}(E),$
where
    $\mu _{\delta }^{h}(E)=\inf \left\{\left.\sum _{i=1}^{\infty }h\left(\mathrm {diam} (C_{i})\right)\right|\mathrm {diam} (C_{i})\leq \delta ,\bigcup _{i=1}^{\infty }C_{i}\supseteq E\right\}.$
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 for t ≥ 0, strictly positive for t > 0, and continuous on the right for all t ≥ 0.

### Packing dimension

Packing dimension is constructed in a very similar way to Hausdorff dimension, except that one "packs" E from inside with pairwise disjoint balls of diameter at most δ. Just as before, one can consider functions h : [0, +∞) → [0, +∞] more general than h(δ) = δ$^s$ and call h an exact dimension function for E if the h-packing measure of E is finite and strictly positive.

## Example

Almost surely, a sample path X of Brownian motion in the Euclidean plane has Hausdorff dimension equal to 2, but the 2-dimensional Hausdorff measure μ$^2$(X) is zero. The exact dimension function h is given by the logarithmic correction
    $h(r)=r^{2}\cdot \log {\frac {1}{r}}\cdot \log \log \log {\frac {1}{r}}.$
I.e., with probability one, 0 < μ$^h$(X) < +∞ for a Brownian path X in R$^2$. For Brownian motion in Euclidean n-space R$^n$ with n ≥ 3, the exact dimension function is
    $h(r)=r^{2}\cdot \log \log {\frac {1}{r}}.$