An abstract Kolmogorov theorem, and an application to metric spaces and topological groups

Using the method of averaging over the supports of $\delta$-functions, we generalise Kolmogorov's fundamental theorem on the existence of a trigonometric Fourier series that diverges almost everywhere to any bounded biorthonormal systems of complex-valued functions on an arbitrary measurable space. The abstract Kolmogorov theorem thus obtained is applied to construct divergent Fourier series on metric spaces and topological groups. We establish the existence of a Fourier series in the system of characters of an arbitrary compact Abelian group that is divergent almost everywhere.
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