R. Henstock. The general theory of integration. The Clarendon Press Oxford University Press, New York, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....28C10; Secondary 26A39. 1 Introduction Through the 20th century serval theories of integration have been developed, for instance, the integral of Riemann, Lebesgue, Denjoy, Perron, McShane, Henstock Kurzweil and other approaches dealing with functions defined on R or more generally on R n (see [2, 4, 5, 7, 8, 11]) Dealing with abstract cases, it is remarkable that a considerable amount of measure theory needs to be developed before that the integral can be studied (see [3, 6, 10, 14] Unlike this conventional way A. Anger and C. Portenier investigated in [1] an abstract theory of integration which is ....

R. Henstock. The general theory of integration. The Clarendon Press Oxford University Press, New York, 1991.

.... [11] However neither the Riemann nor Lebesgue integral obeys the Fundamental Theorem, for example the derivative of the function x 2 sin(1=x 2 ) and 0 at x = 0) is not Lebesgue integrable over [0; 1] We have instead chosen to define the integral in HOL following the KurzweilHenstock theory [9, 10]. This is simpler to define than the Lebesgue integral, and has the Lebesgue integral as a special case. Moreover, it obeys the Fundamental Theorem of Calculus which has been proved in HOL. Hence we are justified in evaluating integrals by letting Maple find an antiderivative. To see how the ....

R. Henstock, The General Theory of Integration, Clarendon Press, Oxford 1991.

....third and final stage to all Lebesgue integrable functions. 6. 3 The Henstock integral Apart from the R integral which we have described above, there are two other notions of generalized Riemann integrals which have developed since the early sixties, namely, the McShane and the Henstock integrals [70, 60]. These are basically integrals for real valued functions on R. Their generalisations to R n also exist but they are more involved. The basic McShane integral is equivalent to the Lebesgue integral with respect to to the Lebesgue measure in the sense that a real valued function is Lebesgue ....

R. Henstock. The General Theory of Integration. Oxford Math. Monographs. Clarendon Press, 1991.

....integral or simply gauge integral . In the following, we give a sketch of the definition of this integral following the terminology given in [18] and note some results that have already been proved in HOL. A fuller introduction may be found in the undergraduate textbook [8] or the definitive [12]. The limiting process involved in the gauge integral seems rather obscure at first sight, but the intuition can be seen quite clearly if we consider integrating a derivative. Suppose f is differentiable for all x lying between a and b. Then given any such x and any ffl 0, we know that there ....

R. Henstock, The General Theory of Integration, Clarendon Press, Oxford 1991.

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