Burkill-Cesari integral

An extension of Burkill's integration process (for interval and rectangular functions; cf. Burkill integral) to set functions in abstract settings, introduced by L. Cesari in [a8], [a9]. It finds applications in the Weierstrass-type approach to the calculus of variations (see [a1], [a10], [a11], the survey [a2], and [a5], [a6], [a7]).

Elementary integration process for interval functions.

Given an interval , let denote the collection of all closed subintervals. Let be the family of finite partitions of and let be the mesh function defined by .

An interval function is said to be integrable if the limit exists and is finite.

One of the main applications of this process is to the Jordan length of a continuous rectifiable curve , which coincides with the integral of the function , .

Burkill integration process for rectangle functions.

Given a closed rectangle , let denote the collection of all closed subrectangles with sides parallel to the axes. Let be the family of Cartesian subdivisions of and let be the mesh function defined by .

A rectangle function is said to be Burkill integrable if the limit

exists and is finite.

It has been proved that the area of a continuous surface of bounded variation coincides with the Burkill integral of a suitable rectangular function.

Burkill–Cesari integration process in an abstract setting.

Given a topological space , let be the family of all subsets of . Let be a given class of sets, called "intervals" . A finite system is a finite collection of non-overlapping intervals, i.e. and , , , where and denote the -interior and -closure, respectively.

Let be a given net of finite systems and let be the function defined by when and otherwise.

A function , where is a Banach space, is said to be Burkill–Cesari integrable over if the limit

exists.

An efficient condition for the existence of the Burkill–Cesari integral is Cesari quasi-additivity: A function is said to be quasi-additive [a8] over if for each there exists a such that for every there exists a such that for every ,

where , .

The function is said to be of bounded variation if .

A quasi-additive function is Burkill–Cesari integrable. Moreover, if is quasi-additive and of bounded variation on , then both functions and are quasi-additive on all subsets .

An analogous Burkill–Cesari weak integration process was introduced in [a3].

Subject to a suitable strengthening of the setting, the Burkill–Cesari integral admits extension to measures. Moreover, the total variation of the Burkill–Cesari measure of a function coincides with the Burkill–Cesari measure of .

Connections of the Burkill–Cesari process with martingale theory were presented in [a4] (cf. also Martingale). A characterization of lower semicontinuity for Burkill–Cesari integrals was proposed in [a7]. The integration processes of Riemann, Lebesgue–Stieltjes, Hellinger, Bochner, Pettis, and Henstock can be regarded as particular Burkill–Cesari integrals (cf. also Riemann integral; Lebesgue–Stieltjes integral; Hellinger integral; Bochner integral; Pettis integral; Kurzweil–Henstock integral).

The variation, length and area of a curve or surface of bounded variation (not necessarily continuous) find a meaningful definition in terms of the Burkill–Cesari integral. Furthermore, a definition of weighted length and area can be introduced by means of this process.

More generally, the Burkill–Cesari integral has important applications in the definition of Weierstrass-type integrals in the calculus of variations (cf. Variational calculus). In fact the classical Lebesgue functionals are valid only with respect to Sobolev's variety, but the corresponding Weierstrass integrals provide a good and meaningful extension to the bounded variation setting.

References

[a1]	J.C. Breckenridge, "Burkill–Cesari integrals of quasi additive interval functions" Pacific J. Math. , 37 (1971) pp. 635–654
[a2]	C. Vinti, "Nonlinear integration and Weierstrass integral over a manifold: connections with theorems on martingales" J. Optimization Th. App. , 41 (1983) pp. 213–237
[a3]	P. Brandi, A. Salvadori, "Sull'integrale debole alla Burkill–Cesari" Atti Sem. Mat. Fis. Univ. Modena , 23 (1978) pp. 14–38
[a4]	P. Brandi, A. Salvadori, "Martingale ed integrale alla Burkill–Cesari" Atti Accad. Naz. Lincei , 67 (1979) pp. 197–203
[a5]	P. Brandi, A. Salvadori, "A quasi-additive type condition and the integral over a BV variety" Pacific J. Math. , 146 (1990) pp. 1–19
[a6]	P. Brandi, A. Salvadori, "On the non-parametric integral over a BV surface" J. Nonlinear Anal. , 13 (1989) pp. 1127–1137
[a7]	P. Brandi, A. Salvadori, "On the semicontinuity of Burkill–Cesari integral" Rend. Circ. Mat. Palermo , 63 (1994) pp. 161–180
[a8]	L. Cesari, "Quasi-additive set functions and the concept of integral over a variety" Trans. Amer. Math. Soc. , 102 (1962) pp. 94–113
[a9]	Cesari,L, "Extension problem for quasi-additive set functions and Radon–Nykodym derivatives" Trans. Amer. Math. Soc. , 102 (1962) pp. 114–145
[a10]	G. Warner, "The Burkill--Cesari integral" Duke Math. J , 35 (1968) pp. 61–78
[a11]	G. Warner, "The generalized Weierstrass-type integral " Ann. Scuola Norm. Sup. Pisa , 22 (1968) pp. 163–191