Let us denote [M.sub.[tau]], by the sigma algebra generated by [tau].

One may now conclude that [M.sub.[sigma]-w] = [M.sub.w], [M.sub.[sigma]-s] = [M.sub.s] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (where [M.sub.a.m] is the sigma algebra generated by the Arens-Mackey topology).

We also denote [B.sub.[tau]], by the sigma algebra generated by [B.sub.[tau]].

(1) The sigma algebra [B.sub.[tau]] is properly contained in [M.sub.[tau]], which makes us face two diagrams [M.sub.[tau]]'s and [B.sub.[tau]]'s.

A

sigma algebra ([sigma]--algebra) F on X is a class of subsets of X such that:

Let E and H denote a pair of real separable Hilbert spaces and {[OMEGA], F, [F.sub.t], t [member of] I, P} a complete filtered probability space with [F.sub.t] [subset] F a family of nondecreasing complete sub-sigma algebras of the sigma algebra F and I = [0, T], T < [infinity].

Let U be a compact Polish space and M(U) the space of finite Borel measures on the sigma algebra B(U) on U.

To be more precise, let [lambda] x P denote the product of Lebesgue measure and the probability measure on the Cartesian product I x [OMEGA], and let P denote the sigma algebra of [G.sup.t]-predictable subsets of the set I x [OMEGA] and [mu] denote the restriction of the measure [lambda] x P on to P.

Some terms: measurable space, probabilistic space, elementary event, the space of elementary events,

sigma algebra, complex event, the function of probability, the probability of events, axioms of probability theory and Borel [sigma]-algebra as defined in (Sarapa, 1993).

To be more precise, let [lambda] x P denote the product of Lebesgue measure and the probability measure on the Cartesian product I x [OMEGA], and let P denote the sigma algebra of [G.sub.t]-predictable subsets of the set I x [OMEGA] and [mu] denote the restriction of the measure [lambda] x P on to P.

It follows from the assumption (A1) and the growth properties in assumptions (A2) and (A3) that the integrands in (4.7) and (4.8), are dominated by the following [mu]-integrable processes {[[zeta].sub.1], [[zeta].sub.2]} given by the conditional expectations (with respect to the current of sigma algebras [G.sub.s] [subset] [F.sub.s], s [member of] I) as shown below,

To consider the stochastic system we need to introduce the probability space with filtration ([OMEGA], F, [F.sub.t[greater than or equal to]0] ,P) where [OMEGA] is the sample space and F is the

sigma algebra of Borel subsets of [OMEGA] and [F.sub.t[greater than or equal to]0] is a nondecreasing family of subsigma algebras of the

sigma algebra F and P is the probability law.