Ž .
convex function, ⌽ k 0, ⌽ 0 s 0. The ⌽-approximation of a real A A-measur-Ž< < . able function f is the process of minimizing H ⌽ f y h d among the real func-⍀ tions h of some class M M. In this paper we consider a class M M of ⌺-measurable functions, where ⌺ ; A A is a -lattice that is totally ordered by a closed subset of w x the interval 0, 1 . A special example of M M is the set of radialᎏnondecreasing or nonincreasingᎏfunctions defined on the unit ball of ޒ n , and where is the Lebesgue measure. The primary purpose of this paper is to show a somewhat strong relation between ⌽-approximation by M M and ⌽-approximation by constants on subsets of ⍀. This assertion is enlightened by the following facts. Best step monotone ⌽-approximants are formed by best constant ⌽-approximants on subsets of ⍀. Furthermore, appropriate sequences of best step monotone ⌽-approximants have subsequences that converge a.e. to a best monotone ⌽-approximant, and conversely, any best monotone ⌽-approximant is a limit of a sequence of best step monotone ⌽-approximants. On the one hand, these facts are applied to give methods for constructing best ⌽-approximants. On the other hand, they yield rather theoretical results when M M is, more generally, the class of ⌺ X -measurable functions, where ⌺ X ; A A is an arbitrary -lattice.