I left off yesterday asking how to measure a surface (something sort of 2-dimensional) in 3-space, and noting that Lebesgue measure would give this set 0 area. This is where Hausdorff measure comes in. There is in fact a continuum of Hausdorff measures, one for each real number. First it is helpful to define the diameter of a set, which is just the farthest distance apart of two points in the set. For a circle, the diameter is the usual definition. Also, we’ll denote the diameter of a set U by |U|.
The construction of the measure is typically described in two steps: Suppose we are trying to find the m-dimensional Hausdorff measure of a set U. Then in the first step, we cover U by sets no larger than some number, d, and look at the sum of the diameters of these sets raised to the mth power. The smallest (well, infimum) of these coverings will be the number we associate with the parameter d. In the second step, we let d go to zero. The Hausdorff measure is the limit, if it exists. In symbols,
, where for each k. Then . There is an easy result (see, for example, Falconer’s “Geometry of Fractal Sets“) that says that for any set U, there is a unique value, called the Hausdorff dimension of U so that and . Note that the Hausdorff measure might be 0 or infinite, but this number will still exist.
We should also note that:
- Nowhere in the definition did I use anything but distance, making all these concepts valid in metric spaces, as well as in Euclidean (vector) spaces.
- Nowhere in the definition did I require the Hausdorff dimension to be an integer.
Expanding on this second point brings us into discussing fractals, which I have discussed a bit in the past week, but have not defined. One reason for this is that I have not seen a great definition. Self similarity is important, as is having a Hausdorff dimension different from its topological dimension. At the very least, any self similar set whose Hausdorff dimension is not an integer would be given “fractal” status, though this would miss many good examples. In any case, for any self similar set, the Hausdorff dimension is log(m)/log(s), where m is the number of copies at each step, and s is the (linear) scaling factor.
So if you view the real plane as a self similar set taken by starting with a square, then dividing it into four squares, and then each of those into four squares, and so on, then the number of copies is 4 and the scaling factor is 2 (Remember, it is the linear scaling factor! Each side is half the length it used to be.), so the Hausdorff dimension is log(4)/log(2), which is precisely 2, since .
I have printed below two more examples of fractals and their dimensions. Thanks to Leonid Kovalev for pointing out how to generate the Sierpinski carpet. Also see this great wikipedia post for a much more complete list of fractals and their dimensions.