Boolean valued topologies and Minlos' theorem
by
Ulrich Hoehle
FB 7 Mathematik, Bergische Universitaet, D-42097 Wuppertal, Germany

A probability \sigma-algebra is a pair (A, \pi) where A is a Boolean \sigma-algebra and \pi is a probability measure defined on A. In particular, the underlying Boolean algebra A of a probability \sigma-algebra is always complete.
An A-valued topological space is a pair (X, \tau) where X is a set and \tau is a subframe of A^X. There exists an adjoint situation between the categories of locales and A-valued topological spaces (cf. U. Höhle, Many valued topology and its applications, Kluwer 2001).
Let L⁰(A, \pi) be the vector space of all \pi-almost everywhere defined, real valued random variables. If A is atomless, then it is well known that there does not exist any ordinary topology O on L⁰(A, \pi) such that pointwise \pi-almost everywhere convergence is equivalent to convergence in the sense of O. But there exists an A-valued topology \tau₀ on L⁰(A, \pi) such that pointwise \pi-almost everywhere convergence is equivalent to convergence in the sense of \tau₀ - i.e. for every j₀ in L⁰(A, \pi) and for every sequence (j_n)_{n in N} in L⁰(A, \pi) the following assertions are equivalent:

1. lim_{n --> \infty} j_n = j₀ \pi-almost everywhere,
2. g(j₀) <= \/ _n=1^\infty( /\ _m=n^\infty g(j_m)) for allg in \tau.

1. E is nuclear.
2. For every linear map l: E --> L⁰(A, \pi) the following assertions are equivalent:

   1. l is continuous w.r.t. the ordinary topology of convergence in measure on L⁰(A, \pi).
   2. l is continuous w.r.t. the A-valued topology \tau₀ on L⁰(A, \pi).

Date received: May 15, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeq-09.