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An extension of Burkill's integration process (for interval and rectangular functions; cf. [[Burkill integral|Burkill integral]]) to set functions in abstract settings, introduced by L. Cesari in [[#References|[a8]]], [[#References|[a9]]]. It finds applications in the Weierstrass-type approach to the calculus of variations (see [[#References|[a1]]], [[#References|[a10]]], [[#References|[a11]]], the survey [[#References|[a2]]], and [[#References|[a5]]], [[#References|[a6]]], [[#References|[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|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|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 [[#References|[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 [[#References|[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 [[#References|[a4]]] (cf. also [[Martingale|Martingale]]). A characterization of lower semicontinuity for Burkill–Cesari integrals was proposed in [[#References|[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|Riemann integral]]; [[Lebesgue–Stieltjes integral|Lebesgue–Stieltjes integral]]; [[Hellinger integral|Hellinger integral]]; [[Bochner integral|Bochner integral]]; [[Pettis integral|Pettis integral]]; [[Kurzweil–Henstock 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|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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.C. Breckenridge,   "Burkill–Cesari integrals of quasi additive interval functions"  ''Pacific J. Math.'' , '''37'''  (1971)  pp. 635–654</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Vinti,   "Nonlinear integration and Weierstrass integral over a manifold: connections with theorems on martingales"  ''J. Optimization Th. App.'' , '''41'''  (1983)  pp. 213–237</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Brandi,   A. Salvadori,   "Sull'integrale debole alla Burkill–Cesari"  ''Atti Sem. Mat. Fis. Univ. Modena'' , '''23'''  (1978)  pp. 14–38</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Brandi,   A. Salvadori,   "Martingale ed integrale alla Burkill–Cesari"  ''Atti Accad. Naz. Lincei'' , '''67'''  (1979)  pp. 197–203</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P. Brandi,   A. Salvadori,   "A quasi-additive type condition and the integral over a BV variety"  ''Pacific J. Math.'' , '''146'''  (1990)  pp. 1–19</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P. Brandi,   A. Salvadori,   "On the non-parametric integral over a BV surface"  ''J. Nonlinear Anal.'' , '''13'''  (1989)  pp. 1127–1137</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  P. Brandi,   A. Salvadori,   "On the semicontinuity of Burkill–Cesari integral"  ''Rend. Circ. Mat. Palermo'' , '''63'''  (1994)  pp. 161–180</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  L. Cesari,   "Quasi-additive set functions and the concept of integral over a variety"  ''Trans. Amer. Math. Soc.'' , '''102'''  (1962)  pp. 94–113</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  Cesari,L,   "Extension problem for quasi-additive set functions and Radon–Nykodym derivatives"  ''Trans. Amer. Math. Soc.'' , '''102'''  (1962)  pp. 114–145</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  G. Warner,   "The Burkill--Cesari integral"  ''Duke Math. J'' , '''35'''  (1968)  pp. 61–78</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  G. Warner,   "The generalized Weierstrass-type integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106064.png" />"  ''Ann. Scuola Norm. Sup. Pisa'' , '''22'''  (1968)  pp. 163–191</TD></TR></table>