On the finest Lebesgue topology on the space of essentially bounded measurable functions

Marian Nowak

Vol. 140 (1989), No. 1, 155–161

DOI: 10.2140/pjm.1989.140.155

Abstract

Let (Ω,Σ,μ) be a σ-finite measure space and let 𝒯₀ and 𝒯_∞ denote the usual metrizable topologies on L⁰ and L^∞, respectively. In this paper the space L^∞ with the mixed topology γ(𝒯_∞,𝒯₀|_L^∞) is examined. It is proved that γ(𝒯_∞,𝒯₀|_L^∞) is the finest Lebesgue topology on L^∞, and that it coincides with the Mackey topology τ(L^∞,L¹).

Mathematical Subject Classification 2000

Primary: 46E10

Secondary: 46E30

Milestones

Received: 15 November 1987

Published: 1 November 1989