The unique break point between the ∞ range and the 0 range is called the Hausdorff dimension of A. Intuitively, if you take a set and try to measure its volume using too small a dimension, you always get ∞, and if you try to measure its volume using too large a dimension, you always get 0. If you play with this definition for a while, what you find out is that for any given set A, there is at most one real number s so that 0 < H s(A) < ∞. For details see Federer, Geometric measure theory, 2.10.] Thus one could carry out Carathéodory's construction using coverings by members of a different family from the family of balls. [Note also that in a metric space one can find the diameter of any set, not only a ball: (*) To prove this, compute the integral of exp(-|x| 2) in polar coordinates in R n, using Fubini's theorem and the fact that Γ(1/2) = π 1/2. Turns out to yield a measure with all the nice properties one wants in a measure, and which agrees with s- dimensional Lebesgue measure for integer s. The constant for balls is 2 -sα(s), where α(s) is a conventional notation for the s-volume of the unit ball of A by balls of maximum diameter t. The key idea in its construction is to notice that, for integer dimensions s and simple sets A (such as an s- dimensional box or ball), the s- dimensional volume of A is a constant multiple of the s-th power of the diameter of A. As the name suggests, H s measures the "s- dimensional volume" of subsets of R n. H s is called s- dimensional Hausdorff measure (usually it's written with a capital script H). For each real number s ≥ 0, there is a measure H s defined for subsets of R n. In more detail: We work throughout in a fixed Euclidean space R n. If you have seen the Koch snowflake, its Hausdorff dimension is log 4/log 3. A shape whose Hausdorff dimension is not an integer is a fractal (the term is due to Benoit Mandelbrot, he of the famous set). A way of assigning a number for " dimension" to shapes which are not as nice as a line or sphere, or more generally a manifold.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |