Vol. 49, No. 1, 1973

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A characterization of the Mackey uniformity m(Lāˆž, L1) for finite measures

Keith Duncan Stroyan

Vol. 49 (1973), No. 1, 223ā€“228

Let μ be a finite positive measure on a σ-algebra 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.

Mathematical Subject Classification 2000
Primary: 46E30
Secondary: 02H25
Received: 21 August 1972
Published: 1 November 1973
Keith Duncan Stroyan