The error in discrete Φ-approximation

Let φ : [0, ∞] → [0, ∞) be a convex function φ(x) > 0, if x > 0, φ(0) = 0 and let (Ω, A, μ) be a measure space, where Ω or a finite subset of N, A = 2Ω, and μ is any finite measure. We consider the set Mφ,W(f) of best monotone φ-approximates of the function f in the Orlicz space Lφ = Lφ(Ω, A, μ). We obtain a best φ-approximant as limit of one sequence of best approximates in finite dimension. Also, we give a computational algorithm to estimate the error in φ-approximation of functions in the subspace L∞φ, by nondecreasing functions.