We construct a candidate for the natural best L1-approximation to an integrable function, ƒ, by elements of an L1-closed convex proximinal set. If ƒ is a Lebesgue integrable function on [0, 1] and the approximating set is the set of all nondecreasing functions, we show that our construction gives an extension of the known natural best L1-approximation operator from ∪p > 1 Lp to L1. In the course of doing this, we also complete the characterization, given in (Huotari, Meyerowitz, and Sheard, J. Approx. Theory 47 (1986), 85–91) of the set of all best L1-approximations. Finally, in the case of isotonic approximation to a function of several variables, we extend a previous result concerning the almost everywhere convergence of the best Lp-approximations, p > 1, to the natural best L1-approximation.

For ƒ in L1[0, 1], let μ1(ƒ|M) be the set of all best L1-approximations to ƒ by nondecreasing functions, let ƒ = inf μ1(ƒ|M) and let f̂ = sup μ1(ƒ|M). We show that μ1(ƒ|M) is compact in L1 and provide a description of the elements of μ1(ƒ|M) and of its extreme points in terms ƒ and f̄f̂