Year: 1970
Type: article
Abstract: In [6] and [7] a constructive definition is given for the Denjoy-Perron integral.In [7] the author remarks, "If, however, we wished to include also discontinuous integrals we should have to modify some details of the definition".Although many discontinuous integrals have been defined this approach does not seem to have been used.In this note it is shown that simple modifications of the definition in [6,7] give a construction that includes many known integrals as special cases. A General Constructive Definition.Unless otherwise stated functions in this paper will either be finite real-valued point functions with domain the fixed bounded non-empty closed interval, 7 0 , of the real line, or finite real-valued interval functions with domain all closed sub-intervals of some sub-interval of 7 0 .The symbols /, J, /', J', etc. will denote closed sub-intervals small letters /, g will usually denote point functions, capital letters F, G interval functions script capitals S > ~C, '/ will denote various integrals.In particular X, (Γ,£D* will denote the Lebesgue, Denjoy-Khintchine and Denjoy-Perron integrals respectively, [6,7] further -£(/, 7) will denote value of the -/^-integral of / on the interval 7 with a similar notation for other integrals.We will distinguish a class K of interval functions, with domain all closed subintervals of 7 0 , and a non-negative function (not necessarily finite) V defined on couples (F, 7), where 7 is in the domain of F. The following assumptions are made about the class S and the function V.(I) If JF'€(E, and G€(5 then F+Gc®.(II) If Fz Gt then F(I) is completely determined by the values of F(J), for Jc7.(VI) Let Q be a closed set with end points those of 7 and contiguous intervals jn 7, {7 n } weiV ; further suppose that for each n, I n is in th^ domain of
Source: Tohoku Mathematical Journal
Author: P. S. Bullen
Institution: University of British Columbia
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W1980043541