Abstrakt
The Doob convergence theorem implies that the set of divergence of any martingale has measure zero. We prove that, conversely, any G(delta sigma) subset of the Cantor space with Lebesgue-measure zero can be represented as the set of divergence of some martingale.
In fact, this is effective and uniform. A consequence of this is that the set of everywhere converging martingales is Pi(1)(1)-complete, in a uniform way.
We derive from this some universal and complete sets for the whole projective hierarchy, via a general method. We provide some other complete sets for the classes Pi(1)(1) and Sigma(1)(2) in the theory of martingales.