Introduction to Hausdorff Measure

It was not until my third year of graduate school that I thought to ask why geometric measure theory was called geometric measure theory.  It just always seemed like the answer should have been fairly obvious- the questions were very geometric, and everyone likes measures, right?  Well the prototypical problem in GMT is Plateau’s problem, which is roughly: given a closed curve in three space, what surface has the smallest area, and the given curve as a boundary.

A curve in 3-space, and a minimizing surface.

In order to study this problem for a “large enough” collection of sets to guarantee a solution, mathematicians could not rely on nice parameterizations existing over which surface integrals would be taken.  This can be thought of in analogy with the fact that “every nth degree polynomial has n roots” is not true over the real numbers, but it is true over the complex numbers.  But the notion of the area of such a “rectifiable set” is not immediately clear.

Covering a set by squares

More precisely, in mathematics, a “measure” is a function which eats sets, and returns a number that is, intuitively, the “size” of that set.  Perhaps the most commonly used measure- Lebesgue measure- is constructed exactly in this intuitive manner.  We first cover the set with squares (or lines in dimension 1, or cubes in dimension 3, or hypercubes in dimension 4, etc) with side length 1, then squares of side length 0.5, then 0.25, and so on.  By adding up the area (the area of an n-cube we define to be the side length to the nth power) of the squares in each case, we get a sequence, and if (when) that sequence converges, we call the number the (Lebesgue) measure of the set.

The problem with this approach is that if I have a reasonable line sitting in the plane, the Lebesgue measure of that line will be 0.  Similarly for a surface sitting in 3-space.  This is, in some sense, good.  The “length” of a point should be 0, and the “area” of a line should also be 0.  But this is a bad thing in the sense that maybe someone (like Joseph Plateau) is interested in the area of a surface in 3-space.

Covering the same set with smaller squares.

Tomorrow will be more on finding the area of such a surface (hint: the answer is Hausdorff measure)!  And fractals!  Then maybe the discussion will come back to “why is it called geometric measure theory?”


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