In this paper we prove the following: let (t) be a continuous function, increasing in [0, ∞) and (+0) = 0. Then there exists a series of the formwith the following property: for each ε > 0 a weighted function (x), 0 < (x) 1, |{x ∈ [0, 2 ] : (x) = 1}| < ε can be constructed, so that the series is universal in the weighted space L 1 [0, 2 ] with respect to rearrangements.

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In this paper, we prove the following: let ω(t) be a continuous function with ω(+0) = 0 and increasing in [0, ∞). Then there exists a series of the formwith the following property: for each ε > 0 a weight function µ(x), 0 < µ(x) ≤ 1, |{x ∈ [0, 1) : µ(x) = 1}| < ε can be constructed so that the series is universal in the weighted space L 1 µ [0, 1) both with respect to rearrangements and subseries.

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