Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Universal and complete sets in martingale theory   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
${\bf\Pi}^1\_1$-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 ${\bf\Pi}^1\_1$ and
${\bf\Sigma}^1\_2$ in the theory of martingales.

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The categories of this paper are: gr-qc
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The categories of this paper are: quant-ph
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The categories of this paper are: supr-con
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The categories of this paper are: cond-mat
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The categories of this paper are: dg-ga
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The categories of this paper are: nlin
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The categories of this paper are: mtrl-th
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The categories of this paper are: adap-org
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The categories of this paper are: ao-sci
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The categories of this paper are: physics
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The categories of this paper are: chem-ph
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The categories of this paper are: q-alg
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The categories of this paper are: hep-th
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The categories of this paper are: bayes-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: econ