#### Publication Date

9-2012

#### Abstract

To distinguish between random and non-random sequence, Kolmogorov and Martin-Lof proposed a new definition of randomness, according to which an object (e.g., a sequence of 0s and 1s) if random if it satisfies all probability laws, i.e., in more precise terms, if it does not belong to any definable set of probability measure 0. This definition reflect the usual physicists' idea that events with probability 0 cannot happen. Physicists -- especially in statistical physics -- often claim a stronger statement: that events with a very small probability cannot happen either. A modification of Kolmogorov-Martin-Lof's (KLM) definition has been proposed to capture this physicists' claim. The problem is that, in contrast to the original KLM definition, the resulting definition of randomness is not uniquely determined by the probability measure: for the same probability measure, we can have several different definitions of randomness. In this paper, we show that while it is not possible to define, e.g., a unique *set* R of random objects, we can define a unique *sequence* R_{n} of such sets (unique in some reasonable sense).

## Comments

Technical Report: UTEP-CS-12-34

Published in

Mathematical Structures and Modeling, 2012, Vol. 26, pp. 49-56.