We will investigate the properties of series of functions which are unconditionally convergent almost everywhere on [0, 1]. We will establish the following theorem: If the series σ k=1 ∞ f k(x) converges unconditionally almost everywhere, then there exists a sequence {Βk} 1 ∞ ,Βk↑ ∞, such that ifλ k ≤Β k , k=1, 2,..., the series σ k=1 ∞ λk/k(x) converges unconditionally almost every-where.