Weak L1-computability and Limit L

Brattka, Miller and Nies [1] showed that some randomness notions are char-acterized by dierentiability of some classes of functions. They also proposed to study which class corresponds to which randomness notion. Pathak, Rojas and Simpson [3] and independently Rute [4] showed that a real in the unit interval is Schnorr random if and only if the Lebesgue dierentiation theorem for the point holds for all eective version of L1-computable functions. Then its other randomness versions are of our interest. The author [2] gave several character-izations of the class of the eective version of L1-computable functions. Then we also study its other randomness versions. Let (X; d; ) be a computable metric space and be a computable measure on it. The following denition and result are by [2]. A integral test for Schnorr randomness is a nonnegative lower semicomputable function f: X! R whose integral is computable. A function f is L1-computable with an eective code if there exists a computable sequence fsng of nite rational step functions such that f = limn sn and jjsn+1 snjj1 2 n for all n. De nition 1 ([2]). Let f: X! R be a function whose domain is the set of Schnorr random points. Then f is L1-computable with an eective code i f is the dierence between two integral tests for Schnorr randomness. The following is the Martin-lof randomness verions of this result. Recall that an integral test is a nonnegative lower semicomputable function t: X! R withR td < 1. De nition 2. A function f: X! R is weakly L1-computable if there exists a computable sequence fsng of nite rational step functions such that f(x) = limn sn(x) an