Nuclear subspace of L0 and the Kernel of a linear measure

Abstract

Let E be a locally convex space. Then E is nuclear metrizable if and only if there exists a σ-additive measure μ on E′ such that L: E → L 0( E′, μ), L( x) = 〈 x, ·〉, is an isomorphism. Let E be quasi-complete or barrelled. Suppose that there exists a σ-additive measure ν on E satisfying ( E′, τ ν )′ ⊃ E. Then E′ b is an isomorphic subspace of L 0( E, ν) and nuclear, where b is the strong dual topology and τ ν is the L 0( E, ν) topology. In the case where E is an LF space, for a random linear functional L: E → L 0(Ω, U , P), the next conditions are equivalent: (a) The cylinder set measure μ on E′ determined by L is σ-additive and (b) x n → 0 in E implies that L( x n ) → 0, P-a.s.