It is important to answer this question because, to any sigma algebra on B(H), a particular type of operator valued measurable functions is corresponded.

m] is the sigma algebra generated by the Arens-Mackey topology).

3) By some calculations, we will find the diagram of the sigma algebra [B.

Let us denote [sigma]-([GAMMA]), by the sigma algebra generated by T.

A

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

t] [subset] F a family of nondecreasing complete sub-sigma algebras of the sigma algebra F and I [equivalent to] [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.

Equivalently, one may consider the measurability with respect to the Borel sigma algebra generated by the weak star open (or closed) subsets of M(U).

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).

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.

The authors discuss a broad range of topics, from the basic concepts of probability to advanced topics for further study, including; integrals, martingales, and

sigma algebras.