Rod Nillsen
Mathematics research
The Carathéodory definition of measurable sets (1994-- )
Introduction
The paper "Irrational rotations motivate measurable sets", ( Elem. Der Math. 56 (2001),1-17) arose from the fact that many applications of analysis assume a prior knowledge of measure theory, a situation which creates a barrier for many undergraduate students, who are thereby prevented by seeing some of the most interesting applications of modern analysis. The results may be viewed as slides of a talk given at a meeting of the Australian Mathematical Society. The main purpose of the paper is to show how Carathédory's condition for a set to be measurable arises naturally from considering a specific problem in ergodic theory. The problem is: given an irrational rotation on the set of complex numbers of modulus 1, calculate the possible values of the outer measure of a subset that is invariant under this rotation. The paper does not presuppose any knowledge of measure theory, since it is concerned with calculating the outer measure of a set, and the notion of outer measure is defined in the paper. The paper also has some discussion on the pedagogical issues arising from the startling quality of Carathédory's original definition, a quality commented on by many mathematicians. Work on this area has continued with Amos Koeller, Peter Nickolas and Graham Williams.
Comments
These comments are an edited version of comments in the paper in Elem. Der Math. They are primarily concerned with pedagogical issues which were an original motivation for the work.
In the general theory of integration, measure theory and, in particular, the notion of a measurable set, play a central role. Various definitions of the notion of a measurable set have been used, but perhaps the most famous is the one given by Carathéodory in 1914. Carathéodory's definition depends on the notion of outer measure Since Carathéodory originally gave his definition of a measurable set, it has frequently been the subject of comment or defence in a way unusual for the definition of mathematical concept. For example, in their book Real and Abstract Analysis , Edwin Hewitt and Karl Stromberg write: ``How Carathéodory came to think of this definition seems mysterious, since it is not in the least intuitive. Carathéodory's definition has many useful implications".
Also, Paul Halmos comments in his book Measure Theory ``It is rather difficult to get an understanding of the meaning of ......... measurability except through familiarity with its implications...The greatest justification of this apparently complicated concept is, however, its possibly surprising but absolutely complete success as a tool in proving the important and useful extension theorem". However, not everyone has accepted that the definition of a measurable set can be justified by appealing to a usefulness which may be at the time quite unclear but which will be amply justified in the future. Writing in his Conjectures and Refutations about Michel Loève's treatment of measurable sets in the latter's work Probability Theory , Imre Lakatos says: "...how on earth can he know which of these most complicated instruments will be needed for the operation? Certainly he already has some idea what he will find and how he will proceed. But why then, this mystical set-up of putting the definition before the proof?"
The objections of Lakatos apply not simply to the definition of a measurable set, but also to the way in which the definitions of many mathematical concepts often are presented. However, in the case of measurable sets, the situation is more acute than in many other cases. The definition of a measurable set in itself is not really counter-intuitive, since the ``counter-intuitive" definition can in fact be given an intuitive interpretation, as Halmos explains in his book on Measure Theory (p.44). However, the definition is counter-intuitive in the sense that the it seems to appear out of nowhere and its immediate intuitive interpretation does not provide a broader context, or even the suggestion of one, in which it might appear as natural, transparent or necessary. The definition of a measurable set appears to be unusually ``remote" from the context which gives it its mathematical importance. On the one hand, this very ``remoteness" may have an intriguing and mysterious quality. Indeed, it can be held that crucial mathematical concepts and definitions retain an intrinsic air of unexpectedness and mystery which no amount of heuristic justification or tracing of historical origins can eliminate. A good definition remains primarily an artistic act. However, on the other hand, the definition of a measurable set may have a seeming artificiality and a lack of any intrinsic indication as to its possible longer-term significance...... So, although the motivation for it was different, the work here can be considered to show how Carathéodory's definition of a measurable set arises from a specific "problem situation", a pedagogical approach recommended by Lakatos.
Rod Nillsen, October 2005