Wednesday, March 6, 2013

Lebesgue Measure II

For the definition of the Lebesgue Measure and some of its simple applications to sets, see the previous post.

So far, many of the sets discussed have been products of non-degenerate intervals (or sets approximated by these products of intervals), and these have had positive Lebesgue Measures. Since an interval in R1 is an uncountable set, so are the products of intervals in Rn. The other sets that have been considered have been either finite or countable, and all of these have had Lebesgue Measure 0. Therefore, thus far, negligibility has coincided with countability. However, there are also uncountable sets which are negligible.

The key concept is that a subset A of Rn is negligible if it has a dimension less than n. Before exploring the implications of this statement, consider a simple example:

Let the set A be interval [0,1] embedded in R2 (illustrated below). This, being an interval, is an uncountable set, but it can be shown by a succession of approximations by squares in R2 that this set has Lebesgue Measure 0.

A portrayal of the set A in R2. The standard Cartesian coordinates and axes are superimposed for clarity. The first approximation (top), uses the single square (2-prism) [0,1]x[0,1], which clearly contains A, to approximate the volume of A. Having side length 1, the volume is 1. In the second approximation (middle), the union of two squares, J21 and J22, each with side length 1/2, clearly contains A, the total volume of the approximation (denoted by J2) being (1/2)2 + (1/2)2 = 1/2. The third approximation (bottom), follows a similar approach, with four squares of side length 1/4 containing A and yielding the approximation J3 = 1/4 to A. Clearly, if the side length of the approximating squares is halved in each successive approximation, the resulting estimated volume is halved each time. Since this can be done indefinitely, λ(A) = 0.

The above example shows how squares, the most basic units of area in R2, can be used to calculate the volumes of other sets, including negligible ones. This example also motivates a useful condition for negligibility: A set S is negligible in Rn if, for every ε > 0, no matter how small, there exists a (finite) collection of rectangular n-prisms Ji, such that S is contained within the union of the Ji and the Lebesgue measure of the union of the Ji is less than or equal to ε.

Since the set A in the example is one-dimensional (in our as yet intuitive sense), its negligibility is consistent with the previous statement that a set of some dimension embedded in a real space of higher dimension is negligible, as 1 < 2. Note that, if the same set A were being considered as a subset of R1, it would not be negligible, but rather λ(A) would equal 1. This reveals the importance of specifying the dimension of the ambient space Rn. More generally, if A is a "well-behaved" manifold of dimension m embedded within Rn, and m < n, then A is negligible under the Lebesgue Measure.

The purpose of the qualifier "well-behaved" is to distinguish between two types of dimension. The common idea of dimension, the number of perpendicular directions one can "contain" in space, is what is called the topological dimension. However, the Lebesgue Measure also depends on something called the Hausdorff dimension, named after the German mathematician Felix Hausdorff. This alternate measure of dimension applies to a larger class of sets, notably fractals, and does not assign only integer values; many sets have Hausdorff dimension between 1 and 2, for example.

For any space, the Hausdorff dimension of the space is always greater than or equal to the topological dimension (if applicable) of the same space. Thus, even if a set topologically has a smaller dimension than the real space around it, it may still have nonzero Lebesgue Measure.

Take, for example, the Cantor Set. It is produced by removing the middle third of the interval [0,1], and, on every subsequent step, removing the middle third from each remaining line segment. After an infinite number of steps, the set remains. The properties of this set are proved elsewhere. The Cantor set is a fractal, an uncountable set, but not a dense set. Also, if its points are expressed in ternary (base 3) notation, the numerical expansion of each member of the set contains only 0's and 2's (in base 3, the possible digits are 0, 1, and 2). The Hausdorff dimension of the Cantor set is related to its property of containing points which "use" only 2 out of the 3 digits of the ternary system; the dimension is ln(2)/ln(3) ≈ .6309. This figure gives some idea to "how big" the set is. Since this is less than 1, the dimension of the real space in which it is embedded, the Cantor Set has Lebesgue Measure 0.

In the above case, the topological dimension can be considered to be 0, as the set contains only "isolated" points. However, the Hausdorff dimension, though greater, was still less than that of the ambient space, giving the same result: the Cantor set is negligible. In other cases, the distinctness of the Hausdorff dimension does affect the Lebesgue Measure. Consider the figure below.

The above illustrates three stages of the construction of a fractal known as a space-filling curve, or Peano curve. It is a fractal because it is self-similar (structures are duplicated on smaller levels in each step), and after infinite steps, it fills every point in the space it occupies, which, for simplicity, we will assume to be the unit square [0,1]x[0,1] in R2. Locally, each part of the space-filling curve looks like a line, giving the set topological dimension 1. However, it fills every point in a two-dimensional region, and has Hausdorff dimension 2. With respect to the Lebesgue Measure on R2, the space-filling curve is not a negligible set, but rather has a positive Lebesgue Measure, equal in magnitude to the volume of the unit square, that is, 1.

This post has illustrated the application of the Lebesgue Measure to an even wider variety of sets. However, there are some even more exceptional sets to which the Lebesgue Measure does not assign a value at all! (see the next post)
Sources: Advanced Calculus of Several Variables by C.H. Edwards Jr., Vitali Sets on Wikipedia

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