Abstract:
Necessary and sufficient conditions for the existence of a universal series in any system of measurable functions are established. It is proved that if there exists a universal series in a system $\Phi$, then there exists a universal series in this system such that, for any measurable function $f(x)$, there exists a subsequence of partial sums $S_{m_k}(x)$ converging to $f(x)$ almost everywhere and such that the upper density of the subsequence of indices $(m_k)_{k=1}^{\infty}$ is $1$. Questions on the density of $(m_k)_{k=1}^{\infty}$ are also examined for general almost everywhere convergent subsequences of measurable functions $(U_{m_k}(x))_{k=1}^{\infty}$.
Bibliography: 7 titles.
Keywords:system of measurable functions, universal series, density of a subsequence of natural numbers, upper density, lower density.
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