Monotone approximation in several variables (Q1072041)

Let \(\Omega\) denote the unit n-cube \([0,1]^ n\), and let M be the set of all real valued functions on \(\Omega\) which are nondecreasing in each variable. If f is a bounded Lebesgue measurable function on \(\Omega\) and \(1<p<\infty\), let \(f_ p\) denote the best \(L_ p\)-approximation to f by elements of M. It is shown that \(f_ p\) converges almost everywhere as p decreases to one to a best \(L_ 1\)-approximation to f by elements of M. If f is continuous, then \(f_ p\) is continuous and converges uniformly as \(p\downarrow 1\) to a best \(L_ 1\)-approximation to f by element of M and \(f_ p\) converges uniformly as \(p\to \infty\) to a best \(L_{\infty}\)-approximation to f by element of M.