Abstract
A complete orthonormal system of functions { Θ n } n=1 ∞, Θ n ∈ L [0,1] ∞, defined on the closed interval [0,1] is constructed such that ∑ n=1 ∞ a n Θ n diverges almost everywhere for any {a n} n=1 ∞∉ l 2 . For the constructed system the following result is true: Corollary 1. Any nontrivial series in the system { Θ n } n=1 ∞ which converges in measure to zero diverges almost everywhere.
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