Completions in measure of languages and related combinatorial problems

We consider measures of languages induced by Bernoulli distributions on the letters of a given alphabet. Of particular interest are languages having a measure equal to 1 with respect to all positive Bernoulli distributions (Bernoulli sets). The main object of the paper is to study conditions ensuring that a given language has a finite Bernoulli completion, i.e., it is included in a finite Bernoulli set. Some characterizations of languages having finite Bernoulli completions are given. In the case of a two-letter alphabet it is shown that one can decide whether a finite language has a finite Bernoulli completion or not. Moreover, any finite code over a two-letter alphabet has a finite Bernoulli completion. Finally, we prove that two finite languages have the same measure with respect to all Bernoulli distributions if and only if each of the two languages can be obtained from the other by using a finite number of times three suitable measure-invariant transformations.