# Borel measure

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## Borel measure

[bə′rel ‚mezh·ər]
(mathematics)
A measure defined on the class of all Borel sets of a topological space such that the measure of any compact set is finite.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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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.
(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  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 ) 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].

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