## Comparison between different types of abstract integrals in Riesz spaces.(English)Zbl 0881.28008

Let $$E$$ be a Dedekind complete Riesz space and $$[a,b]$$ a real interval. In the first part of the paper, the authors give a characterization of “Riemann-integrable” functions $$f:[a,b]\to E$$ well known in the real-valued case. In the second part, they introduce an integral for real-valued functions with respect to an $$E$$-valued measure: A function $$f$$ is integrable if there is an $$L^1$$-Cauchy sequence of simple functions $$(o)$$-converging in measure to $$f$$. This concept of integrability is compared with several other concepts of integrability.
Reviewer: H.Weber (Udine)

### MSC:

 28B15 Set functions, measures and integrals with values in ordered spaces
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### References:

 [1] Aguilera N. E., Harboure E. O.,On the search for weighted norm inequalities for the Fourier transform, Pacific J. Maths,104 (1) (1983), 1–14. · Zbl 0527.42001 [2] Betancor J.J.,A mixed Parseval’s equation and a generalized Hankel transformation of distributions, Canad. J. Math.,41 (1989), 274–284. · Zbl 0666.46046 [3] Betancor J.J.,Two complex variants of a Hankel type transformation of generalized functions, Port. Mathematica,46 (3) (1989), 229–243. · Zbl 0736.46035 [4] Erdelyi A.,On some functional transformations, Rend. Sem. Mat. Univ. Pol. Torino,10 (1950/51), 217–234. [5] Erdelyi A., Magnus W., Oberhettinger F., Tricomi F. G.,Tables of integral transforms, II, Mc Graw Hill Book Company, New York, Toronto, London, 1954. · Zbl 0055.36401 [6] Hayek N.,Estudio de la ecuación diferencial xy”+(v+1)y’+y=0 y de sus aplicaciones, Collectanea Mathematica,18 (1966/67), 55–174. [7] Lee W. Y.,On Schwartz’s Hankel transformation of certain spaces of distributions, SIAM J. Math. Anal.,6 (2) (1975), 427–432. · Zbl 0293.46028 [8] Méndez J. M. R., Socas M.M.,A pair of generalized Hankel-Clifford transformations and their applications, J. Math. Anal. Appl.,154 (2) (1991), 543–557. · Zbl 0746.46031 [9] Rooney P. G.,On the range of certain fractional integrals, Can. J. Math.,24 (1972), 1198–1216. · Zbl 0249.44009 [10] Rooney P. G.,On the representation of functions as Fourier transforms, Canad. J. Math.,11 (1959), 168–174. · Zbl 0086.08701 [11] Rooney P. G.,On an inversion operator for the Fourier transformation, Canad. Math. Bull.,3 (2) (1960), 157–165. · Zbl 0097.31102 [12] Rooney P. G.,On the representation of functions by the Hankel and some related transformations, to appear in Proc. Roy. Soc. Edimburgh. · Zbl 0837.44005 [13] Schwartz L.,Théorie des distributiones, Hermann, Paris, 1957. [14] Slater L. J.,Confluent hypergeometric functions, Cambridge University Press, Cambridge, 1939. · Zbl 0141.07203 [15] Stein E. M., Weiss G.,Fourier analysis on euclidean spaces, Princeton University Press, Princeton, New Jersey, 1975. · Zbl 0232.42007 [16] Watson G. N.,A treatise on the theory of Bessel functions, Cambridge University Press, London, 1958. · Zbl 0083.20702 [17] Widder D. V.,The Laplace transform, Princeton University Press, Princeton, 1946. · Zbl 0060.24801 [18] Zemanian A. H.,Inversion formulas for the distributional Laplace transformation, SIAM J. Appl. Math.,14 (1966), 159–166. · Zbl 0147.11904 [19] Zemanian A. H.,Generalized integral transformations, Interscience, New York, 1968. (Reprinted by Dover, N.Y., 1987). · Zbl 0181.12701
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