A

Borel measure defined on the space X is a non-negative function [mu]: Bo(X) [right arrow] R with the following properties:

where [sigma] is a complex

Borel measure of bounded variation on [L.sub.2][0, T], f [member of] [L.sub.p]([R.sup.r]) with 1 [less than or equal to] p [less than or equal to] [infinity], and {[v.sub.1], ..., [v.sub.r]} is an orthonormal subset of L2[0, T], He then investigated relationships between the conditional Fourier-Feynman transforms and the conditional convolution products of the functions given by (3).

Therefore, we may assign a graph [GAMMA]([mu]) = (V([mu]), E([mu])) to any probability

Borel measure [mu] on X.

Let [mu] be a [sigma]-finite

Borel measure on a Polish space S.

(ii) Every real finitely additive finite

Borel measure on a Borel space X, regarded as a continuous functional on [L.sup.[infinity]](X), attains its norm.

By the Riesz representation theorem, for every continuous linear functional u on [C.sub.0]([OMEGA] x [OMEGA]'), there is a

Borel measure [mu] on [OMEGA] x [OMEGA]' such that <u, x> = [[integral].sub.[OMEGA] x [OMEGA]'] x d[mu] on [C.sub.0]([OMEGA] x [OMEGA]') and [parallel]u[parallel] = [absolute value of [mu]] ([OMEGA] x [OMEGA]').

Let [mu] be a finite and positive

Borel measure on the unit circle T = {z [member of] C, [absolute value of z] = 1}.

Our proof is obtained by the regularity properties of the corresponding regular

Borel measure on [??] and is very different from the given in [3] and other known proofs.

Let [mu] be a countably additive, regular, vector valued,

Borel measure on R taking values in [B *.sub.s].

The

Borel measure [mu] which supported by K is defined by

Shamir (see [8]) have had the natural idea to substitute in the relation (1.1) the combination u[??]v by certain finite

Borel measure [mu] on [??] and to investigate the limit, as [epsilon] [right arrow] [0.sup.+] and [rho] [right arrow] +[infinity], of

We introduce one notation on

Borel measures: Let [mu] be a

Borel measure on [R.sup.D], and let a [member of] [R.sup.D].