An infinite binary sequence x is defined to be strongly useful if there exists a recursive function t, such that for every decidable sequence y there exists a Turing machine using oracle x that decides the nth bit of y within time t(n). Weakly useful sequences x are defined similarly, except that the collection of sequences reducible to x as described only has to form a set that is not of measure 0. In this context, sets of measure 0 (small sets) are defined in a special way, using martingales. Some related but weaker properties, whose definitions are too long for this review, are also considered. Among them are weak and strong depths, which have already been considered in earlier papers, as well as computably weak depth, which is newly introduced. It is shown that every weakly useful sequence is computably weakly deep.

The main result of the paper asserts that there exists a sequence that is weakly useful, but not strongly useful. The proof uses an extension of a technique known as martingale diagonalization, and is quite intricate.