This is Martin-Löf's original definition. For a finite binary string w we let Cw denote the cylinder generated by w. This is the set of all infinite sequences beginning with w, which is a basic open set in Cantor space. The product measure μ(Cw) of the cylinder generated by w is defined to be 2−|w|. Every open subset of Cantor space is the union of a countable sequence of disjoint basic open sets, and the measure of an open set is the sum of the measures of any such sequence. An effective open set is an open set that is the union of the sequence of basic open sets determined by a recursively enumerable sequence of binary strings. A constructive null cover or effective measure 0 set is a recursively enumerable sequence of effective open sets such that and for each natural number i. Every effective null cover determines a set of measure 0, namely the intersection of the sets . A sequence is defined to be Martin-Löf random if it is not contained in any set determined by a constructive null cover. Constructive martingales (Schnorr 1971): A martingale is a function such that, for all finite strings w, , where is the concatenation of the strings a and b. This is called the "fairness condition": if a martingale is viewed as a betting strategy, then the above condition requires that the bettor plays against fair odds. A martingale d is said to succeed on a sequence S if where is the first n bits of S. A martingale d is constructive (also known as weakly computable, lower semi-computable) if there exists a computable function such that, for all finite binary strings w for all positive integers t, A sequence is Martin-Löf random if and only if no constructive martingale succeeds on it. Interpretations of the definitions The Kolmogorov complexity characterization conveys the intuition that a random sequence is incompressible: no prefix can be produced by a program much shorter than the prefix. The null cover characterization conveys the intuition that a random real number should not have any property that is "uncommon". Each measure 0 set can be thought of as an uncommon property. It is not possible for a sequence to lie in no measure 0 sets, because each one-point set has measure 0. Martin-Löf's idea was to limit the definition to measure 0 sets that are effectively