Almost Everywhere Convergence of Greedy Algorithm with Respect to Vilenkin System

In this paper, we prove that for any ε ∈ (0, 1) there exists ameasurable set E ∈ [0, 1) with measure |E| > 1 ? ε 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).