ERRETT A. BISHOP It turns out to be possible to extend I to L in such a way that 1 ' (X, L1 , I) also satisfy the axioms, and in addition L1 is complete under the metric p(f,g) = ~ ( l f - g l 1. The only real problem in recovering the classical Daniel1 theory is posed by the classical result that if f E L1 then the set A = {x EX: f(x) kt] t is integrable for all t 0 (in the sense that its characteristic function x t ' defined by x (x) = 1 if f k ) 2 t and xtk) = 0 if f(x) t , is in L1). The t constructive version is that A is integrable for all except countably many t t 7 0. The proof of this requires a rather complex theory, called the theory of profiles. Y. K. Chan informs me that he has been able to simplify the theory of profiles considerably. He has also effected a considerable simplifi- cation in another trouble-spot of [5], the proof that a non-negative linear functional I on the set L = C(X) of continuous functions on a compact space X satisfies the axioms for an integration space presented above. (Axiom (2) is the troublemaker. ) Constructive integration theory affords the point of departure for some recent constructivizations of parts of probability theory. There is no (constructive) way to prove even the simplest cases of the ergodic theorem, such that if T denotes rotation of a circle X through an angle a, then for each integrable function f : X + R and almost all x in X, the averages converge. (The difficulty comes about because we are unable to decide for instance whether a = 0.) One way to recover the essence of the ergodic theorem constructively, 'and in fact deepen it considerably, is to show that the sequence If 3 satisfies certain integral inequalities, analogous to the up- N crossing inequalities (see [14]) of martingale theory. This was done in the context of the Chacon-Ornstein ergodic theorem in [l], and even more gener- ally in [3] . John Nuber [2 31 takes another route, He presents sufficient conditions, close to being necessary, that the sequence IfN 1 actually converges a. e., in the context of the classical Birkhoff ergodic theorem. More recently, in an unpublished manuscript, he has generalized his conditions to the context of the classical Chacon-Ornstein theorem.
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