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A problem in two-dimensional integration

Part of: Real functions

Published online by Cambridge University Press:  09 April 2009

Ralph Henstock
Ralph Henstock
Affiliation:
School of Physical Sciences (Mathematics)The New University of UlsterColeraine Northern, Ireland
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Abstract

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If two functions of a real variable are integrable over two intervals, say of t, τ, respectively, then the product of the two functions should be integrable over the rectangular product of the two intervals of t and τ. For the Lebesgue integral, definable using non-negative functions alone, the proof is easy. For non-absolute integrals such as the Perron, Çesàro-Perron, and Marcinkiewicz-Zygmund integrals we have difficulties since the functions cannot be assumed non-negative. But the present paper gives a proof.

MSC classification

Secondary: 26A39: Denjoy and Perron integrals, other special integrals
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 35 , Issue 3 , December 1983 , pp. 386 - 404
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Bauer, H., ‘Der Perronsche Integralbegriff und seine Beziehung zum Lebesgueschen’, Monatsh. Math. Phys. 26 (1915), 153–198.CrossRefGoogle Scholar
[2]Bosanquet, L. S., ‘A property of Çesàro-Perron integrals’, Proc. Edinburgh Math. Soc. (2) 6 (1940), 160–165, MR 2, 131.CrossRefGoogle Scholar
[3]Bullen, P. S., ‘The Pn-integral’, J. Austral. Math. Soc. 14 (1972), 219–236, MR 47, 8783.CrossRefGoogle Scholar
[4]Bullen, P. S. and Lee, C. M., ‘On the integrals of Perron type’, Trans. Amer. Math. Soc. 182 (1973), 481–501, MR 49, 3057.CrossRefGoogle Scholar
[5]Bullen, P. S. and Lee, C. M., ‘The SCnP-integral and the P n+1-integral’, Canad. J. Math. 25 (1973), 1274–1284, MR 53, 13489.CrossRefGoogle Scholar
[6]Burkill, J. C., ‘The approximately continuous Perron integral’, Math. Z. 34 (1931), 270–278, Zbl 2.386.CrossRefGoogle Scholar
[7]Burkill, J. C., ‘The Çesàro-Perron integral’, Proc. London Math. Soc. (2) 34 (1932), 314–322, Zbl 5.392.CrossRefGoogle Scholar
[8]Burkill, J. C., ‘The Çesàro-Perron scale of integration’, Proc. London Math. Soc. (2) 39 (1935), 541–552, Zbl 12.204.CrossRefGoogle Scholar
[9]Burkill, J. C., ‘Fractional orders of integrability’, J. London Math. Soc. 11 (1936), 220–226, Zbl 14.258.CrossRefGoogle Scholar
[10]Burkill, J. C., ‘Integrals and trigonometric series’, Proc. London Math. Soc. (3) 1 (1951), 46–57, MR 13, 126.CrossRefGoogle Scholar
[11]Cross, G. E., ‘The relation between two definite integrals’, Proc. Amer. Math. Soc. 11 (1960), 578–579, MR 22, 8094.CrossRefGoogle Scholar
[12]Cross, G. E., ‘The relation between two symmetric integrals’, Proc. Amer. Math. Soc. 14 (1963), 185–190, MR 26, 281.CrossRefGoogle Scholar
[13]Cross, G. E., ‘On the generality of the AP-integral’, Canad. J. Math. 23 (1971), 557–561, MR 44, 378.CrossRefGoogle Scholar
[14]Cross, G. E., ‘The P n-integral’, Canad. Math. Bull. 18 (1975), 493–497, MR 53, 3224.CrossRefGoogle Scholar
[15]Cross, G. E., ‘Additivity of the P n-integral’, Canad. J. Math. 30 (1978), 783–796, MR 58, 6108.CrossRefGoogle Scholar
[16]Ellis, H. W., ‘Mean-continuous integrals’, Canad. J. Math. 1 (1949), 113–124, MR 10, 520.CrossRefGoogle Scholar
[17]Foglio, S., ‘Absolute N-integration’, J. London Math. Soc. 38 (1963), 87–88, MR 26, 3868.CrossRefGoogle Scholar
[18]Foglio, S. and Henstock, R., ‘The N-variational integral and the Schwartz distributions-III’, J. London Math. Soc. (2) 6 (1973), 693–700, MR 48, 2755.CrossRefGoogle Scholar
[19]Gordon, L., ‘Perron's integral for derivatives in L'’, Studia Math. 28 (1967), 295–316, MR 36, 322.CrossRefGoogle Scholar
[20]Hake, H., ‘Über de la Vallée Poussins Ober- und Unterfunktionen einfacher Integrale und die Integraldefinition von Perron’, Math. Ann. 83 (1921), 119–142.CrossRefGoogle Scholar
[21]Henstock, R., ‘A new descriptive definition of the Ward integral’, J. London Math. Soc. 35 (1960), 43–48, MR 22, 1648.CrossRefGoogle Scholar
[22]Henstock, R., ‘The use of convergence factors in Ward integration’, Proc. London Math. Soc. (3) 10 (1960), 107–121, MR 22, 12197.CrossRefGoogle Scholar
[23]Henstock, R., ‘The equivalence of generalized forms of the Ward, variational, Denjoy-Stieltjes, and Perron-Stieltjes integrals’, Proc. London Math. Soc. (3) 10 (1960), 281–303, MR 22, 12198.CrossRefGoogle Scholar
[24]Henstock, R., ‘N-variation and N-variational integrals of set functions’, Proc. London Math. Soc. (3) 11 (1961), 109–133, MR 23 A995.CrossRefGoogle Scholar
[25]Henstock, R., ‘Definitions of Riemann type of the variational integrals’, Proc. London Math. Soc. (3) 11 (1961), 402–418, MR 24, A1994.CrossRefGoogle Scholar
[26]Henstock, R., ‘Majorants in variational integration’, Canad. J. Math. 18 (1966), 49–74, MR 32, 2545.CrossRefGoogle Scholar
[27]Henstock, R., Linear analysis (Butterworths, London, 1968), MR 54, 7725.Google Scholar
[28]Henstock, R., ‘Integration, variation and differentiation in division spaces’, Proc. Roy. Irish Acad. Sect. A 78 (1978), 69–85, MR 80d: 26011.Google Scholar
[29]Henstock, R., ‘The variation on the real line’, Proc. Roy. Irish Acad. Sect. A 79 (1979), 1–10, MR 81d: 26005.Google Scholar
[30]James, R. D., ‘Generalized nth primitives’, Trans. Amer. Math. Soc. 76 (1954), 149–176, MR 15, 611.Google Scholar
[31]James, R. D., ‘Summable trigonometric series’, Pacific J. Math. 6 (1956), 99–110, MR 17, 1198.CrossRefGoogle Scholar
[32]Jeffery, R. L. and Miller, D. S., ‘Convergence factors for generalized integrals’, Duke Math. J. 12 (1945), 127–142, MR 6, 204.CrossRefGoogle Scholar
[33]Looman, H., ‘Uber die Perronsche Integraldefinition’, Math. Ann. 93 (1925), 153–156.CrossRefGoogle Scholar
[34]Marcinkiewicz, J. and Zygmund, A., ‘On the differentiability of functions and summability of trigonometrical series’, Fund. Math. 26 (1936), 1–43, Zbl 14.111.CrossRefGoogle Scholar
[35]McGrotty, J. J., ‘A theorem on complete sets’, J. London Math. Soc. 37 (1962), 338–340, MR 25, 4052.CrossRefGoogle Scholar
[36]McKean, H. P. Jr, Stochastic integrals (Academic Press, New York, 1969), MR 40, 947.Google Scholar
[37]McShane, E. J., Integration (Princeton Math. Series 7, 1944, Princeton Univ. Press), MR 6, 43.Google Scholar
[38]Mukhopadhyay, S. N., ‘On the regularity of the P n-integral and its application to summable trigonometric series’, Pacific J. Math. 55 (1974), 233–247, MR 51, 10546.CrossRefGoogle Scholar
[39]Perron, O., ‘Ueber den Integralbegriff’, Sitzungsber. Heidelb. Akad. Wiss. Math.-Natur. Kl. 16 (1914), 1–16.Google Scholar
[40]Pfeffer, W. F., ‘The Perron integral in topological spaces’, (Russian), Časopis Pěst. Mat. 88 (1963), 322–348, MR 32, 2546.CrossRefGoogle Scholar
[41]Pfeffer, W. F., ‘A definition of the integral in topological spaces’, Časopis Pěst. Mat. 89 (1964), 1 129–147, II 257–277, MR 32, 1318.CrossRefGoogle Scholar
[42]Saks, S., Theory of the integral (2nd English edition, Warsaw, 1937), Zbl 17.300.Google Scholar
[43]Sargent, W. L. C., ‘On a sufficient condition for a function integrable in the Çesàro-Perron sense to be monotonic’, Quart. J. Math. Oxford Ser. 12 (1941), 148–153, MR 3, 228.CrossRefGoogle Scholar
[44]Sarkhel, D. N., ‘A criterion for Perron integrability’, Proc. Amer. Math. Soc. 71 (1978), 109–112, MR 58, 17006.CrossRefGoogle Scholar
[45]Taylor, S. J., ‘An integral of Perron's type defined with the help of trigonometric series’, Quart. J. Math. Oxford Ser. (2) 6 (1955), 255–274, MR 19, 255.CrossRefGoogle Scholar
[46]Thomson, B. S., ‘Constructive definitions for non-absolutely convergent integrals’, Proc. London Math. Soc. (3) 20 (1970), 699–716, MR 42, 3248.CrossRefGoogle Scholar
[47]Verblunsky, S., ‘On a descriptive definition of Çesàro-Perron integrals’, J. London Math. Soc. (2) 3 (1971), 326–333, MR 44, 4161.CrossRefGoogle Scholar
[48]Verblunsky, S., ‘On the Peano derivatives’, Proc. London Math. Soc. (3) 22 (1971), 313–324, MR 44, 2896.CrossRefGoogle Scholar
[49]Ward, A. J., ‘The Perron-Stieltjes integral’, Math. Z. 41 (1936), 578–604, Zbl 14.397.CrossRefGoogle Scholar