How to Measure Volume with a Thread

1. A NAIVE QUESTION. Has it ever occurred to you that one could use a thread to measure volume? Or area? We do not mean imprecise guesses of the following sort: “This parcel must be horribly large, as 23.5 yards of cord were needed to wrap it up.” The answer “Sure, I can use a thread to measure the length and width of my bedroom; it is a rectangle 15 feet by 16 feet, so its area is 240 square feet,” is also unsatisfactory. What we mean is a sort of black box that helps you to determine the volume of every reasonable object once you know the length of a certain piece of thread. Here is the idea. Imagine an ideal, perfectly elastic thread, say of unstretched length one, imprinted with a scale to measure length. Imagine further that this thread is very densely packed into a unit cube C—some Brave Little Tailor, much braver than the Brothers Grimm hero, stretches it (in the process distorting the length scale), pulls it, and twists it so that finally the thread passes through every tiniest speck of the cube, through every cubic millimeter. You can measure the volume of a subset A of C as follows: color all points of A (and hence all points of the thread that lie in A) with a red dye, unpack the thread from the cube, and use the scale to record the total length (we stress: this means total original length, before stretching!) of the colored red pieces of the thread. This total length is equal to the volume of A! No matter where the set A lies in the cube, no matter what shape it has! This sounds almost magical, yet it is possible, at least in an abstract sense. Before you start questioning our sanity, let us abandon our analogy and formulate our question in purely mathematical terms, just to avoid misunderstanding and misinterpretation. We ask: Does there exist a continuous, one-to-one mapping φ: [0, 1) → (0, 1)n (this φ gives the rule for packing the thread into the cube; the condition that φ be one-to-one is quite natural—after all, a thread should have no self-intersections) with the following crucial property: for each Borel subset A of (0, 1)n the one-dimensional measure of its preimage φ−1(A) should be equal to the Lebesgue measure of A (i.e., standard volume for n = 3). (In the previous paragraph A was described as a set whose points were stained with “red dye,” and the Lebesgue measure of the preimage φ−1(A) corresponded, of course, to “total original length of the colored red pieces of the thread before stretching.”)