Henceforth, unless specified otherwise, A is supposed to be a scalar type spectral operator in a complex Banach space (X, [parallel]*[parallel]) and [E.sub.A](*) is supposed to be its strongly o-additive spectral measure (the resolution of the identity) assigning to each

Borel set [delta] of the complex plane C a projection operator [E.sub.A] ([delta]) on X and having the operator's spectrum [sigma](A) as its support [6, 7].

[subset] X : A is a

Borel set such that [lambda](A [intersection] (z + Rx))

Assume that [f.sup.(m)] exists outside a [lambda]- null

Borel set [B.sub.x] [subset or equal to] [a, x] ([lambda] is the Lebesgue measure) such that

For most problems related to finding a reasonable definition for randomness it is sufficient to assume that [OMEGA] = [R.sup.N] is the sample space of real valued sequences with the metric d([omega], [omega]') = [[SIGMA].sub.n] [2.sup.-n] min([absolute value of ([omega](n) - [omega]'(n))], 1), the shifting transformation [theta]([omega])(n) = [omega](n + 1), and the sequence of processes Xn(w) = w(n) and with a probability that makes [X.sub.n] independent with the same distribution [mathematical expression not reproducible] defined as [mathematical expression not reproducible] for all F [member of] B(R)

Borel set.

That is, for any

Borel set B of real numbers, the set

kentucky derby star

Borel set for dubai debut in hot competition

For a

Borel set b' [??]R, set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Propose 4.[10][11][12] Let A be a

Borel set of [summation].

The next question we turn to is whether the distal flows form a

Borel set. To make this question precise, we must consider the distal flows as points in a separable metric space.

Henceforth, unless specified otherwise, A is supposed to be a scalar type spectral operator in a complex Banach space (X, [parallel]*[parallel]) with strongly o-additive spectral measure (the resolution of the identity) [E.sub.A](*) assigning to each

Borel set [delta] of the complex plane C a projection operator [E.sub.A]([delta]) on X and having the operator's spectrum [sigma](A) as its support [7, 8].

They imply for every

Borel set E [subset] [OMEGA] that

Also, from the second main theorem for [f.sub.1] and [a.sub.1], ..., [a.sub.4], we may assume that there exists a

Borel set I [subset] [1, + [infinity]) whose measure [absolute value of I] = + [infinity] and that