## Modular approximation by a filtered family of sublinear operators.(English)Zbl 0636.46025

Let ($$\Omega$$,$$\Sigma$$,$$\mu)$$ be a $$\sigma$$-finite measure space, let $${\mathcal X}$$ be the space of extended real-valued $$\Sigma$$-measurable $$\mu$$-a.e. measurable functions on $$\Omega$$, and let $$\rho$$ : $$\Omega\times {\mathcal X}\to (0,\infty)$$ be a modular function which is measurable with respect to the variable in $$\Omega$$ for each function in $${\mathcal X}$$ and is such that for x,y in $${\mathcal X}$$, $$| x(s)| \leq | y(s)|$$ implies that $$\rho$$ (t,x)$$\leq \rho (t,y)$$ for $$\mu$$- almost all t in $$\Omega$$. Then, X, $$X_{\rho_ S}$$ and $$\rho_ S$$ are defined by
(i) $$\rho_ S(x)=\int_{\Omega}\rho (t,x)d\mu (t);$$
(ii) $$X=\{x\in {\mathcal X}$$, $$\rho$$ (t,ax)$$\to 0$$ a.e. on $$\Omega$$ as $$a\to 0\};$$
(iii) $$X_{\rho_ S}=\{x\in X:\rho_ S(ax)\to 0$$ as $$a\to 0\}.$$
If V is an abstract set, $${\mathcal V}$$ is a filter of subsets of V, $$g: V\to R$$, then g(v)$$\to 0$$ if for every $$\epsilon >0$$, there is a set $$V_ 0\in {\mathcal V}$$ such that $$| g(v)| <\epsilon$$ for $$v\in V_ 0.$$
The main results of this paper relate to classes of q-sublinear or sublinear operators of the form $$\{T_ v$$, $$v\in V\}$$ which are $${\mathcal V}$$-bounded so that there are positive constants $$k_ 1,k_ 2$$ and a function $$g: V\to R_+$$, g(v)$$\to 0$$, and for x,y in $$X_{\rho_ S}$$ there is a set $$V_{x,y}\in {\mathcal V}$$ such that $\rho_ S(a(T_ vx- T_ vy))\leq k_ 1\rho_ S(ak_ 2(x-y))+g(v),\quad for\quad v\in V_{x,y},\quad a>0.$ It is indicated that if $$\nu =0$$ when operators are q-sublinear and $$\nu =-1$$ when operators are sublinear, $$S_ 0\subseteq X_{\rho_ S}$$, S is the set of finite linear combinations of elements of $$S_ 0$$, with closure $$\bar S,$$ and if $$\rho_ S(a(T_ vx+\nu x))\to 0$$ for $$x\in S_ 0$$, $$a>0$$, then for $$x\in \bar S$$ there is a number $$b>0$$ such that $$\rho_ S(b(T_ vx+\nu x))\to 0$$. Applications relating to approximation theory are stated.
Reviewer: G.O.Okikiolu

### MSC:

 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

### Keywords:

sublinear operators; approximation