Best $L_ 1$-approximation of $L_ 1$-approximately continuous functions on $(0,1)^ n$ by nondecreasing functions

Abstract:

For $n \geq 1$, let $\Omega$ denote the open unit $n$-cube, ${(0,1)^n}$. Let $\mu$ denote Lebesgue measure, let $\Sigma$ consist of the Lebesgue measurable subsets of $\Omega$, and let ${L_1} = {L_1}(\Omega ,\Sigma ,\mu )$. Let ${A_1}$ consist of the approximately continuous functions in ${L_1}$ and let $M$ consist of the equivalence classes in ${L_1}$ which contain nondecreasing functions. Let $f \in {A_1}$. It is shown that there is a unique best ${L_1}$-approximation $[g]$ to $f$ in $M$. If $n = 1,g$ is continuous, but if $n > 1,[g]$ may not consist of a continuous function.