A Study of Riesz Space-Valued Non-additive Measures

Many facts in the theory of general Riesz spaces are easily verified by thinking in terms of spaces of functions. A proof via this insight is said to use representation theory. In recent years a growing number of authors has successfully been trying to bypass representation theorems, judging them to be extraneous. (See, for instance, [9,10].) In spite of the positive aspects of these efforts the following can be said. Firstly, avoiding representation theory does not always make the facts transparent. Reading the more cumbersome constructions and procedures inside the Riesz space itself one feels the need for a pictorial representation with functions, and one suspects the author himself of secret heretical thoughts. Secondly, the direct method leads to repeating constructions of the same nature over and over again.

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A Fubini Theorem in Riesz spaces for the Kurzweil-Henstock Integral

Journal of Function Spaces and Applications ◽

10.1155/2011/158412 ◽

2011 ◽

Vol 9 (3) ◽

pp. 283-304 ◽

Cited By ~ 4

Author(s):

A. Boccuto ◽

D. Candeloro ◽

A. R. Sambucini

Keyword(s):

Type Theorem ◽

Riesz Space ◽

Fubini Theorem ◽

Real Plane ◽

Henstock Integral ◽

Riesz Spaces

A Fubini-type theorem is proved, for the Kurzweil-Henstock integral of Riesz-space-valued functions defined on (not necessarily bounded) subrectangles of the “extended” real plane.

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On regular and sigma-smooth two valued measures and lattice generated topologies

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171293000031 ◽

1993 ◽

Vol 16 (1) ◽

pp. 33-40 ◽

Cited By ~ 1

Author(s):

Robert W. Shutz

Keyword(s):

Finitely Additive Measures ◽

Additive Measures

LetXbe an abstract set andLa lattice of subsets ofX.I(L)denotes the non-trivial zero one valued finitely additive measures onA(L), the algebra generated byL, andIR(L)those elements ofI(L)that areL-regular. It is known thatI(L)=IR(L)if and only ifLis an algebra. We first give several new proofs of this fact and a number of characterizations of this in topologicial terms.Next we consider,I(σ*,L)the elements ofI(L)that areσ-smooth onL, andIR(σ,L)those elements ofI(σ*,L)that areL-regular. We then obtain necessary and sufficent conditions forI(σ*,L)=IR(σ,L), and in particuliar ,we obtain conditions in terms of topologicial demands on associated Wallman spaces of the lattice.

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