All Sampling Methods Produce Outliers

Given a computable probability measure P over natural numbers or infinite binary sequences, there is no computable, randomized method that can produce an arbitrarily large sample such that none of its members are outliers of P. In addition, given a binary predicate gamma, the length of the smallest program that computes a complete extension of gamma is less than the size of the domain of gamma plus the amount of information that gamma has with the halting sequence.

Automated summary

There is no computable, randomized method to produce a sample that does not contain outliers of a given computable probability measure P. Additionally, the minimum length of a program that computes a complete extension of a binary predicate gamma is bounded by the size of the domain of gamma and the information it has with the halting sequence.