The space 𝐿^{𝜔} and convex topological rings
Richard F. Arens · Bulletin of the American Mathematical Society · 1946
Introduction.The motive for investigating the class L w of functions belonging to all Z>-classes has no measure-theoretic origin: it was our desire to discover whether or not in every convex metric ring 1 R one could find a system { U] of convex neighborhoods of 0 having the property that/, g E U implies f g G U. We show here that Z, w has no proper convex open set U containing 0 and satisfying the relation UUQ U, thus supplying the desired counter-example.The significance of neighborhood systems of the type {If} described above is made somewhat clearer by a proof that they insure the existence and continuity of entire functions (for example, the exponential function) on the topological ring R.Such neighborhood systems { U} are always present in rings of continuous real-valued functions over any space, provided that convergence means uniform convergence on compact sets.We also consider the relation of L°°, L°, and the 2>-classes, since L u does not seem ever to have been discussed as a topological and algebraic entity.2. Notation and elementary facts.Let us consider measurable functions defined on [O, l].For ^1 we shall consistently employ the usual notation