How to Define Measure of Borel Sets

Introduction In 12] is presented a constructive deenition of Borel subsets of Cantor space and their inclusion relation. They are seen as symbolic expressions built from the Boolean algebra of simple (closed and open) subsets of by formal countable disjunction and conjunction. If X is such a symbolic expression, it is clear for instance how to deene by induction the formal complement X 0 of X: Classically one can think of symbolic expressions as sets of points, and deene the inclusion relation extensionally. Constructively, it is still possible to deene X Y for X; Y Borel subsets without mentioning points, and this is done in 12] using a suitable innnitary one-sided sequent calculus. Using this approach the law X X 0 for instance can be justiied constructively. A theory of Lebesgue measure on is also presented in 12], starting from a measure (b) 2 0; 1] of simple (closed and open) subsets. Constructively, if we want the measure of a subset to be a computable real, we cannot deene in general the measure of even open sets, and the question of measure of Borel subsets is usually not addressed. Here we deene the measure of a Borel set as a bounded \hyperarithmetical real", i.e. a real built from rationals by repeated (may be transsnitely) sups and infs of bounded sequences. The starting point is to follow Borel 3] and try to deene (X) by induction on the construction of X: One insight is that it is more elegant to deene by induction the function X : b 7 ?! (X \ b) where b denotes an arbitrary simple set of Cantor space. It follows indeed from the work of Riesz 13] that X can be deened by structural induction on X, that is, if X n are the components of X, X is a function of Xn while (X) is not a priori a function of (X n): This development is done in an intuitionistic framework using as primitive the notion of generalised inductive deenition. Thus this work provides a solution to what Lusin called \le probl eme de la mesure de M. E. Borel" 10]. It can also be seen as a possible approach to constructive probability theory, and we illustrate this in the last section with a presentation of Borel's normal number theorem. We found it convenient to formulate this approach in the framework of the theory of …