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
where
"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 dimH("E"). Intuitively speaking, "μ""s"("E") = +∞ for "s" < dimH("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" > dimH("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
where
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
I.e., with probability one, 0 < "μ""h"("X") < +∞ for a Brownian path "X" in R2. For Brownian motion in Euclidean "n"-space R"n" with "n" ≥ 3, the exact dimension function is