# 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.

How to Cite This Entry:
Burkill-Cesari integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Burkill-Cesari_integral&oldid=46175
This article was adapted from an original article by P. Brandi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article