ABSTRACT
In this section we try to generalize some results of § 3 in the framework of so-called ideal spaces (Banach lattices) of measurable functions. Examples of ideal spaces are the Lebesgue spaces considered in the last paragraph, Orlicz spaces (which are useful in problems with strong nonlinearities), or Lorentz and Marcinkiewicz spaces (which arise in interpolation theory for linear operators). 4.1. Ideal spaces Many of the results presented in § 3 carryover from Lebesgue spaces to larger classes of spaces which frequently occur in applications. In this subsection we develop the notions which will be needed to formulate the corresponding results. Let n be a nonempty set with a O'-algebra !! of subsets of n (called "measurable sets") and a O'-finite O'-additive measure JJ on !!. Since in aJ1 applications which follow 0 will be a bounded domain in Euclidean space, we shall assume that JJ(O) < 00. We also suppose throughout that the measure JJ is atom-free, although many of the reults which follow also hold for measures with atoms. By 6 = 6(0) we denote the set of all JJ-measurable functions x : o ..... m. equipped with the usual algebraic operations and the metrie
(4.1) p(x, y):= inf {h +JJ( {s : s E 0, Ix(s) - y(s)1 > h})} O<h<oo
or the matrie A f Ix(s) -y(s)1 p(x, y):= 1n 1+ Ix(s) _ y(s)1 dJJ(s).
