Let μ be a finite positive
measure on a σ-algebra ℳ over a set X. As usual L∞(μ) denotes the space of
μ-essentially bounded measurable functions and L1(μ) denotes the space of
μ-integrable functions. In this article we use nonstandard analysis to give a simple
description of the Mackey uniformity m(L∞,L1). The Mackey uniformity is the finest
locally convex linear uniformity on L∞ for which each continuous linear
functional has an L1 representation. The famous theorem of Mackey-Arens
says it is given by uniform convergence on the weakly compact subsets of
L1.
Our description is simply this: Let p be a seminorm on L∞. Then p is Mackey
continuous if and only if whenever g is a finitely bounded element of the nonstandard
extension ∗L∞ which is infinitesimal, except possibly on a set of infinitesimal internal
measure, then p(g) is infinitesimal.
