The Size of Things

In Euclidean geometry, all lengths, areas and volumes are relative. Once a unit of length is chosen, all other lengths are given in terms of this unit. Classical geometry could determine the lengths of straight lines, the areas of polygons and the volumes of simple solids. However, the lengths of curved lines, areas bounded by curves and volumes with curved surfaces were mostly beyond the scope of Euclid. Only a few volumes — for example, the sphere, cylinder and cone — could be measured using classical methods.

Galileo’s one-to-one map between two circles.

Galileo noticed something remarkable about lines of different lengths: the points of a line segment can be put in one-to-one correspondence with the points of another segment of different length. This is most easily shown for two concentric circles of different radii: each radius cuts the two circles at a single point, thereby providing a precise 1-to-1 mapping between the points of the two circles. Informally, we might say that the two circles have “the same number of points”. More formally, and following Cantor, we say that the two sets of points of the circles have the same cardinality.

It is clear that cardinality is a `blunt instrument’: two line segments of different lengths have the same `size’ in terms of it. Moreover, an infinite line, a plane and even a space of countably infinite dimension, all have the same cardinality. Clearly, this has very little to do with the usual concept of size in terms of spacial extent. More refined measure of size is required.

From Calculus to Measure Theory

For the simple sets, we have the geometric length, area and volume. But how can we establish these dimensions for complicated curves, areas and volumes. Integral calculus provided a powerful tool for answering such questions. The length {L} of a curve {y=y(x)} between two limits {x=x_1} and {x=x_2} is

\displaystyle L = \int_{x_1}^{x_2} \sqrt{\mathrm{dx}^2+\mathrm{dy}^2} = \int_{x_1}^{x_2} \sqrt{1+(\mathrm{dy}/\mathrm{dx})^2} \mathrm{dx}

and the area between the curve and the {x}-axis is

\displaystyle A = \int_{x_1}^{x_2} y(x) \mathrm{dx} \,.

The usual definition of an integral, following Bernhard Riemann, works fine for reasonably well-behaved functions over finite intervals. But what if the function to be integrated is wild in some way, or the region over which it is to be integrated is complicated?

Measure theory, initially formulated by Émile Borel and Henri Lebesgue, was developed to answer these questions. The set of functions that are Lebesgue integrable is substantially greater than the set of Riemann integrable functions. For example, the set of rational numbers in the interval {[0,1]} has Lebesgue measure 0; the set of irrational numbers in this interval has measure 1. Thus, informally, there are “many more” irrationals than rationals. This concurs with the cardinality of these sets. However, this concurrence is not guaranteed in general.

While the scope of measure theory is much broader than Riemann’s integral, there are still functions that cannot be integrated and sets that cannot be measured.

Cantor’s Ternary Set

{Seven stages in the construction of Cantor’s set (image from Wikimedia Commons).

Cantor’s ternary set is an example of a set that is uncountable (it has the same cardinality as the real numbers) but that has (Lebesgue) measure zero.To construct the ternary set we proceed iteratively. Starting with the unit interval {C_0 = [0,1]}, we remove the open interval {I_1 = (\frac{1}{3},\frac{2}{3})} corresponding to the “middle third” to get {C_1 = [0,\frac{1}{3}]\cup[\frac{2}{3},1]}. Next, we remove the open middle third of each remaining interval, {I_2 = (\frac{1}{9},\frac{2}{9})\cup(\frac{7}{9},\frac{8}{9})} to get {C_2}, a union of four closed intervals. Continuing this process, we arrive ultimately at the ternary set

\displaystyle C = \bigcap_{n=0}^{\infty} C_n = [0,1] - \bigcup_{n=1}^{\infty} I_n \,.

The set {C} has a number of very interesting properties. {C} is “large”: it is uncountable, making it large compared to the rational numbers. {C} is also “small”: it cannot have a positive length: at each stage, the length is decreased by a factor {\frac{2}{3}} and the limit of {(2/3)^n} is zero. {C} is fractal, with a fractal dimension of {(\log 2 /\log 3 )}. {C} is nowhere dense: the interior of the closure is empty. {C} is totally disconnected. These properties are demonstrated in standard texts on point set topology.

The measure of the set {C} is easily obtained. The length removed at stage {n} is {2^{n-1}/3^n}. Summing these, the total length removed is 1. This implies that the remaining length is 0. The set {C} is of Lebesgue measure zero.

We can show that Cantor’s set has the same cardinality as the real numbers. For any point {x\in C}, we construct a binary number as follows: if {x} is to the left of a middle third removed at stage {n} the {n}th digit is 0. If to the right, the {n}th digit is 1. Clearly this gives a one-to-one correspondence between {C} and all binary numbers in {[0,1]}, so {C} must be uncountable.

Another way to look at this is to note that every point {x} in {C} may be expanded in base 3 as

\displaystyle x = \sum_{n=1}^\infty \frac{t_n}{3^n} \qquad\mbox{where}\qquad t_n\in \{0,2\} \,.

Now we map {x} to the number {y\in[0,1]} with binary expansion

\displaystyle y = \sum_{n=1}^\infty \frac{b_n}{3^n} \qquad\mbox{where}\qquad b_n\in \{0,1\}

with {b_n = t_n/2}. It is clear that the binary expansion of every rational in the unit interval is obtainable in this way (we omit a technical detail that ensures uniqueness of these expansions). We thus have a one-to-one correspondence between the Cantor ternary set {C} and the unit interval, so both must have the same cardinality.

Thus {C}, which has measure zero and is nowhere dense, is also uncountable. It is both big and small. This is a salutary reminder that intuition can be a fickle guide.

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That’s Maths II: A Ton of Wonders

by Peter Lynch now available.
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