SCHIZOPHRENIA IN CONTEMPORARY MATHEMATICS inherently inconsistent and to familiarize oneself with the dangers of self- reference. Not only is there insufficient time, but I would not be competent to re- view all of the recent advances of constructive mathematics, including those of ad hoc constructivism as well as those taking place under constructivist philosophies at variance with those that I have presented here, for example the recursivist constructivism of Markov and his school in Russia. (I have been told that some of the recent advances in differential equatibns have tended to present that subject in a more constructive light. Perhaps Felix Browder will give us some information about those developments. ) I shall restrict my- self in what remains to selected developments with which I am familiar, that seem to me to be of special interest. Brouwer [6] was the first to develop a constructive theory of measure and integration, and the intuitionist tradition (see [19] and [31] for instance) in Holland carried the development further, working with Lebesgue measure on ELn . In [I] I worked with arbitrary measures (both positive and negative) on locally compact spaces, recovering much of the classical theory. The Daniel1 integral was developed in full generality in [5]. The concept of an integration space postulates a set X, a linear subset L of the set of all partially-defined functions from X to R, and a linear functional I from L to IR having the properties (1) if f E L, then If1 E L and min[f,l] E L (2) if f E L and f E L for each n, such that fn 2 0 and aD n C I(fn) converges to a sum that is less than I(f), then n= 1 Z- f (x) converges and is less than fk), for some x in the common n=l n domain of f and the functions f n (3) I(p) f 0 for some p E L - 1 (4) lim I(min {f,nI) = I(f) and lirn I(min[:lfl, n 1) = O for n-) OJ n-, all f in L. We define L1 to consist of all partially defined functions f from X to !R such that there exists a sequence ff 1 of elements of L such that n w w m (a) znzl I( Ifnl converges and (b) 2 f k) = f k ) whenever ZnZl lfn(x)l n=l n converges.
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