In this paper, we prove that for any e ∈ (0, 1) there exists ameasurable set E ∈ [0, 1) with measure |E| > 1 − e such that for any function f ∈ L1[0, 1), it is possible to construct a function $$\tilde f \in {L^1}[0,1]$$ coinciding with f on E and satisfying $$\int_0^1 {|\tilde f(x) - f(x)|dx < \varepsilon } $$ , such that both the Fourier series and the greedy algorithm of $$\tilde f$$ with respect to a bounded Vilenkin system are almost everywhere convergent on [0, 1).