Nuclear subspace of L0 and the Kernel of a linear measure

Let E be a locally convex space. Then E is nuclear metrizable if and only if there exists a [sigma]-additive measure [mu] on E' such that L: E --> L0(E', [mu]), L(x) = <x, ·>, is an isomorphism. Let E be quasi-complete or barrelled. Suppose that there exists a [sigma]-additive measure [nu] on E satisfying (E', [tau][nu])' [superset or implies] E. Then E'b is an isomorphic subspace of L0(E, [nu]) and nuclear, where b is the strong dual topology and [tau][nu] is the L0(E, [nu]) topology. In the case where E is an LF space, for a random linear functional L: E --> L0([Omega], , P), the next conditions are equivalent: (a) The cylinder set measure [mu] on E' determined by L is [sigma]-additive and (b) xn --> 0 in E implies that L(xn) --> 0, P-a.s.