A constructive minimal integral which includes Lebesgue integrable functions and derivatives. (English) Zbl 0980.26006
One of the more interesting developments in integration theory has been the discovery of integrals. Let us call them d-integrals, that integrate all Lebesgue integrable functions and all derivatives, but which are strictly less general than the Denjoy-Perron integral, the classical integral that has these properties. Such a d-integral, the minimal d-integral, is the Bruckner-Fleissner-Foran (BFF-) integral [A. M. Bruckner, R. J. Fleissner and J. Foran, Colloq. Math. 50, 289-293 (1986; Zbl 0604.26006)],that is based on the simple observation that such a minimal d-integral \(F\) of \(f\) must have the properties: (i) \(F' = f\) almost everywhere, (ii) for some differentiable \(H\), \(F-H\) is AC.
In an earlier paper the first author gave a Riemann (R) definition, of a d-integral, the C-integral, in a simple and ingenious way [B. Bongiorno, Matematiche 51, No. 2, 200-313 (1996; Zbl 0929.26007)]. A function \(f\) is R-integrable with integral \(S\) if for all \(\varepsilon>0\) there is a positive function \(\delta\) such that \(\big|\sum_{i=1}^n f(x_i)|I_i|- S\big|< \varepsilon\) whenever the partition \(\{I_i, x_i\), \(1\leq i\leq n\}\) is \( \delta\)-fine, that is \(I_i\subset ]x_i- \delta(x_i),x_i+ \delta(x_i) [\), \(1\leq i\leq n\). This integral is the Lebesgue integral, usually called the McShane integral when defined this way. If we require \(x_i\in I_i\), \(1\leq i\leq n\), then the integral is the more general Denjoy-Perron integral, usually called the Henstock-Kurzweil integral if defined this way. The C-integral is obtained if we require instead that \(\sum_{i=1}^n \rho(I_i, x_i)< 1/ \varepsilon\).
The object of the present paper is to show that the C-integral is the BFF-integral. For this the essential step is to show that if \(f\) is C- integrable then there is a derivative \(h\) such that \(f- h\) is Lebesgue integrable. The proof is based on methods used in the book by Saks, considering the family of regular intervals, those on which such an \(h\) exists, and then showing that the family of these intervals satisfies Romanovskij’s lemma. The details are technical and difficult.
In an earlier paper the first author gave a Riemann (R) definition, of a d-integral, the C-integral, in a simple and ingenious way [B. Bongiorno, Matematiche 51, No. 2, 200-313 (1996; Zbl 0929.26007)]. A function \(f\) is R-integrable with integral \(S\) if for all \(\varepsilon>0\) there is a positive function \(\delta\) such that \(\big|\sum_{i=1}^n f(x_i)|I_i|- S\big|< \varepsilon\) whenever the partition \(\{I_i, x_i\), \(1\leq i\leq n\}\) is \( \delta\)-fine, that is \(I_i\subset ]x_i- \delta(x_i),x_i+ \delta(x_i) [\), \(1\leq i\leq n\). This integral is the Lebesgue integral, usually called the McShane integral when defined this way. If we require \(x_i\in I_i\), \(1\leq i\leq n\), then the integral is the more general Denjoy-Perron integral, usually called the Henstock-Kurzweil integral if defined this way. The C-integral is obtained if we require instead that \(\sum_{i=1}^n \rho(I_i, x_i)< 1/ \varepsilon\).
The object of the present paper is to show that the C-integral is the BFF-integral. For this the essential step is to show that if \(f\) is C- integrable then there is a derivative \(h\) such that \(f- h\) is Lebesgue integrable. The proof is based on methods used in the book by Saks, considering the family of regular intervals, those on which such an \(h\) exists, and then showing that the family of these intervals satisfies Romanovskij’s lemma. The details are technical and difficult.
Reviewer: Peter S.Bullen (Vancouver)
