Generalized Orlicz Spaces. Locally Solid Group Topologies

Abstract

0. S a & g r o u p of Lo and t a group topology on S.Notation. I f ( f a ) is a CATJCHY net in (S, t) converging to f in (Lo, t o ) ,we say that ( f n ) determines f . For Y t S, let Y* be the set of all f E Lo such that there is a net in Y determining f . Put L, : = S*.One immediately verifies the following properties of the *-operator; (1.1.1) Lemma. (a) M H M* is monotone. ( b ) M c M * = -( --M ) * f m M c S . (c) M: + Jf: c ( M , + H2)* for M,, M2 c 8. (1.1.2) Proposition. (a) The system {U*: U is a 0-neighbourhd in (8, t)} form a 0-(b) I f ( f a ) is a net i n 5 determining f E Lo, then f n +f(tl), i.e. (fa) converges to f with (c) M* c z r a V r l (= closure of M with respecf to the supremum of to and z, ) for M c S. Proof. (a) follows from (1.1.1) and the usual characterization of group topologies by O-neighbourhood bases. (b) Let U be a O-neighbourhood in (S,t.), y an index such that f af s E U for a, /I 2 y . If (f.) determines f , then (fafp)ahv determines ffa, hence ff a E U* for /I 2 y. neighbourhood h e for a group topology z, on Lo. respect to tl. (c) follows from (b).( 1.1.3) Corollary. (a) L1 is an open subgtoup oj (LO, t t ) .(b) L, = 5 ' 1 = Bz*Vrl. Proof. Since S ie a subgroup, also L, = S* is a subgroup of Lo by (1.1.1). Since S is a O-neighbourhood in (AS, t), the group L, is a O-neighbourhood in (Lo, q), h_ence q-clopen. Using S c S*, the z,-closedneas of S* and (1.1.2) (c) you get srt c S* c Sr*Vra c PI, hence (b).In the special case considered at the beginning of this section, the spaces L,, S, L, correspond to the spaces of all measurable, of all p-integrable simple, of all p-integrable functions and the topologies to, z, r, to the topology of convergence in measure, the ! I Il,-topology on S and the 11 Il,-topology on Lo, respectively.In our general setting, the topology zl is not always an extension of t ; we have:(1.1.4) Proposition. (a) tl I S 5 t, i.e. the topology on S induced by ti &'coarser than t .

Rearrangement and Convergence in Spaces of Measurable Functions

“…In this section, as a particular case of [6] (see also [13]), we consider Orlicz spaces L N of finitely additive extended real-valued set functions defined on algebras of sets. The space L N has been introduced in [6] in the same way as Dunford and Schwartz [9, page 112] define the space of integrable functions and the integral for integrable functions, and generalize the Orlicz spaces of σ-additive measures defined on σ-algebras of sets.…”