Non-measurable set

In mathematics, a non-measurable set is a special kind of group of numbers that cannot be given a clear size or "volume." This idea helps us understand more about how we measure things like length, area, and volume in advanced math. In a system called Zermelo–Fraenkel set theory, a rule known as the axiom of choice shows that these tricky non-measurable sets do exist.

The idea of non-measurable sets has caused a lot of discussion among mathematicians since it was first introduced. Because of this, famous mathematicians like Borel and Kolmogorov created rules for probability theory that only use sets that can be measured. These sets include all the usual shapes and numbers we work with in everyday math, but proving they can be measured can be quite complex.

In 1970, a mathematician named Robert M. Solovay showed that it is possible, within a special framework called the Solovay model, for all groups of real numbers to be measurable. But his work depends on a special kind of very large number, called an inaccessible cardinal, which cannot be proven to exist using the usual rules of math.

Historical constructions

The first sign that measuring the length of some special sets might be tricky came from Vitali's theorem. Later, a similar idea was shared in American Mathematical Monthly.

We usually think that the size of two separate groups together should just be the added sizes of the two groups. This simple idea is called finitely additive. While this works well for most ideas about area, it is not enough for probability. For more complex situations, we need something called countable additivity.

In some ways, flat surfaces are like lines; we can measure them in a way that stays the same even if we move them around. But for objects with more dimensions, things get stranger. The Hausdorff paradox and Banach–Tarski paradox show that a round ball can sometimes be split into pieces and rearranged into two balls of the same size.

Examples

Imagine a circle where we look at all the points on it. We can turn the circle by special angles that are rational multiples of π. These turns create many small groups of points.

Using a rule called the axiom of choice, we can pick one point from each of these small groups. This creates a special set of points. This set is special because it cannot be given a meaningful "size" or "volume", which makes it a non-measurable set.

Another way to find such sets uses the real numbers divided by the rational numbers.

Consistent definitions of measure and probability

The Banach–Tarski paradox shows that figuring out how to measure volume in three dimensions is tricky. To do this, we have to make one of five important choices:

1. Allow the volume of a set to change when it is rotated.
2. Accept that the volume of two separate sets put together might not add up to the sum of their individual volumes.
3. Label some sets as "non-measurable," meaning we can’t assign them a clear volume.
4. Change the basic rules of Zermelo–Fraenkel set theory with the axiom of choice.
5. Accept that the volume of a special shape might be zero or infinite.

Usually, mathematicians pick the third option. They define a group of sets that can be measured, and most sets used in math fall into this group. It’s often easy to show that a specific shape in a plane can be measured.

In 1970, a mathematician named Solovay showed that we can’t prove the existence of non-measurable sets using just Zermelo–Fraenkel set theory without adding another rule, like the axiom of choice. This was done by creating a special model where every set can be measured, even though not all choices can be made freely.

The axiom of choice connects to many areas of math, like point-set topology and ring theory. But other rules, like determinacy and dependent choice, work well for studying shapes, patterns, and waves, while ensuring all sets can be measured.