May 17, 2007

Referee's Report on:

    Almost Everywhere Domination and Superhighness,
    by Stephen G. Simpson

The paper is a selfcontained survey on recent work on measure based
domination properties in computability theory. Almost everywhere
domination was originally introduced by Dobrinen and the author who
applied this concept to reverse mathematics. More recent work by
various researchers demonstrated the importance of this domination
property for the current work in the theory of algorithmic
randomness. In particular, close relations to central concepts like
K-triviality and lowness for randomness were discovered and helped to
clarify these important notions.

The survey gives an excellent approach to these new exciting
results. For a reader familiar with the basic concepts from
computability theory the survey is selfcontained. Starting with a
short introduction of the basic notions from algorithmic randomness it
leads up to the most recent results. All proofs are given. The clear
style and the unified approach make the very well and carefully
written paper very enjoyable. I assume that it will become the
standard reference for the topic treated. I should add that the paper
does not only present the results in the literature but that it also
introduces some new results and proofs.  

Summarizing, I strongly recommend to accept this excellent paper.