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 $ [ a,b ] \subset \mathbf R $, let $ \{ I \} $ denote the collection of all closed subintervals. Let $ {\mathcal D} _ {[ a,b ] } $ be the family of finite partitions $ D = \{ x _ {0} = a,x _ {1} \dots x _ {n - 1 } , x _ {n} = b \} = \{ I _ {i} \} $ of $ [ a,b ] $ and let $ \delta : {\mathcal D} \rightarrow {\mathbf R ^ {+} } $ be the mesh function defined by $ \delta ( D ) = \max _ {I \in D } | I | $.

An interval function $ \phi : {\{ I \} } \rightarrow \mathbf R $ is said to be integrable if the limit $ {\lim\limits } _ {\delta ( D ) \rightarrow0 } \sum _ {I \in D } \phi ( I ) $ exists and is finite.

One of the main applications of this process is to the Jordan length of a continuous rectifiable curve $ x : {[ a,b ] } \rightarrow \mathbf R $, which coincides with the integral of the function $ \phi ( I ) = \sqrt {| I | ^ {2} + [ x ( \beta ) - x ( \alpha ) ] ^ {2} } $, $ I = [ \alpha, \beta ] $.

Burkill integration process for rectangle functions.

Given a closed rectangle $ [ a,b ] \times [ c,d ] \subset \mathbf R ^ {2} $, let $ \{ R \} $ denote the collection of all closed subrectangles with sides parallel to the axes. Let $ {\mathcal D} = {\mathcal D} _ {[ a,b ] } \times {\mathcal D} _ {[ c,d ] } $ be the family of Cartesian subdivisions $ D = [ R ] $ of $ [ a,b ] \times [ c,d ] $ and let $ \delta : {\{ R \} } \rightarrow \mathbf R $ be the mesh function defined by $ \delta ( D ) = \max _ {R \in D } { \mathop{\rm diam} } ( R ) $.

A rectangle function $ \phi : {\{ R \} } \rightarrow \mathbf R $ is said to be Burkill integrable if the limit

$$ {\lim\limits } _ {\delta ( D ) \rightarrow 0 } \sum _ {R \in D } \phi ( R ) $$

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 $ ( A, {\mathcal G} ) $, let $ {\mathcal M} $ be the family of all subsets of $ A $. Let $ \{ I \} \subset {\mathcal M} $ be a given class of sets, called "intervals" . A finite system $ D = [ I _ {1} \dots I _ {n} ] $ is a finite collection of non-overlapping intervals, i.e. $ I _ {i} ^ {0} \neq \emptyset $ and $ I _ {i} ^ {0} \cup {\overline{I}\; } _ {j} = \emptyset $, $ i \neq j $, $ i,j = 1 \dots n $, where $ I ^ {0} $ and $ {\overline{I}\; } $ denote the $ {\mathcal G} $- interior and $ {\mathcal G} $- closure, respectively.

Let $ ( D _ {t} ) _ {t \in ( T, \gg ) } $ be a given net of finite systems and let $ s : { {\mathcal M} \times {\mathcal M} } \rightarrow {\{ 0,1 \} } $ be the function defined by $ s ( H,K ) = 1 $ when $ H \subset K $ and $ s ( H,K ) = 0 $ otherwise.

A function $ \phi : {\{ I \} } \rightarrow E $, where $ E $ is a Banach space, is said to be Burkill–Cesari integrable over $ M \in {\mathcal M} $ if the limit

$$ {\lim\limits } _ { T } \sum _ {I \in D _ {t} } s ( I,M ) \phi ( I ) = { \mathop{\rm BC} } \int\limits _ { M } \phi $$

exists.

An efficient condition for the existence of the Burkill–Cesari integral is Cesari quasi-additivity: A function $ \phi $ is said to be quasi-additive [a8] over $ M $ if for each $ \epsilon > 0 $ there exists a $ t _ {1} \in T $ such that for every $ t _ {0} \gg t _ {1} $ there exists a $ t _ {2} \in T $ such that for every $ t \gg t _ {2} $,

$$ \sum _ { J } s ( J,M ) \left \| {\sum _ { I } s ( I,J ) \phi ( I ) - \phi ( J ) } \right \| < \epsilon, $$

$$ \sum _ { I } s ( I,M ) \left [ 1 - \sum _ { J } s ( I,J ) s ( J,M ) \right ] \left \| {\phi ( I ) } \right \| < \epsilon, $$

where $ D _ {t _ {0} } = [ J ] $, $ D _ {t} = [ I ] $.

The function $ \phi $ is said to be of bounded variation if $ {\lim\limits \sup } _ {T} \sum _ {I \in D _ {t} } \| {\phi ( I ) } \| < + \infty $.

A quasi-additive function is Burkill–Cesari integrable. Moreover, if $ \phi $ is quasi-additive and of bounded variation on $ A $, then both functions $ \phi $ and $ \| \phi \| $ are quasi-additive on all subsets $ M \in {\mathcal M} $.

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 $ \phi $ coincides with the Burkill–Cesari measure of $ \| \phi \| $.

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