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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]]]).
 
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.==
 
==Elementary integration process for interval functions.==
Given an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b1110601.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b1110602.png" /> denote the collection of all closed subintervals. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b1110603.png" /> be the family of finite partitions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b1110604.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b1110605.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b1110606.png" /> be the mesh function defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b1110607.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b1110608.png" /> is said to be integrable if the limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b1110609.png" /> exists and is finite.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106010.png" />, which coincides with the integral of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106012.png" />.
+
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.==
 
==Burkill integration process for rectangle functions.==
Given a closed rectangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106013.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106014.png" /> denote the collection of all closed subrectangles with sides parallel to the axes. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106015.png" /> be the family of Cartesian subdivisions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106017.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106018.png" /> be the mesh function defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106019.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106020.png" /> is said to be Burkill integrable if the limit
+
A rectangle function $  \phi : {\{ R \} } \rightarrow \mathbf R $
 +
is said to be Burkill integrable if the limit
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106021.png" /></td> </tr></table>
+
$$
 +
{\lim\limits } _ {\delta ( D ) \rightarrow 0 } \sum _ {R \in D } \phi ( R )
 +
$$
  
 
exists and is finite.
 
exists and is finite.
Line 20: Line 51:
  
 
==Burkill–Cesari integration process in an abstract setting.==
 
==Burkill–Cesari integration process in an abstract setting.==
Given a [[Topological space|topological space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106022.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106023.png" /> be the family of all subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106024.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106025.png" /> be a given class of sets, called  "intervals" . A finite system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106026.png" /> is a finite collection of non-overlapping intervals, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106030.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106032.png" /> denote the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106033.png" />-interior and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106034.png" />-closure, respectively.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106035.png" /> be a given net of finite systems and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106036.png" /> be the function defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106037.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106039.png" /> otherwise.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106040.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106041.png" /> is a [[Banach space|Banach space]], is said to be Burkill–Cesari integrable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106042.png" /> if the limit
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106043.png" /></td> </tr></table>
+
$$
 +
{\lim\limits } _ { T } \sum _ {I \in D _ {t} } s ( I,M ) \phi ( I ) = { \mathop{\rm BC} } \int\limits _ { M } \phi
 +
$$
  
 
exists.
 
exists.
  
An efficient condition for the existence of the Burkill–Cesari integral is Cesari quasi-additivity: A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106044.png" /> is said to be quasi-additive [[#References|[a8]]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106045.png" /> if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106046.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106047.png" /> such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106048.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106049.png" /> such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106050.png" />,
+
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} $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106051.png" /></td> </tr></table>
+
$$
 +
\sum _ { J } s ( J,M ) \left \| {\sum _ { I } s ( I,J ) \phi ( I ) - \phi ( J ) } \right \| < \epsilon,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106052.png" /></td> </tr></table>
+
$$
 +
\sum _ { I } s ( I,M ) \left [ 1 - \sum _ { J } s ( I,J ) s ( J,M ) \right ] \left \| {\phi ( I ) } \right \| < \epsilon,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106054.png" />.
+
where $  D _ {t _ {0}  } = [ J ] $,  
 +
$  D _ {t} = [ I ] $.
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106055.png" /> is said to be of bounded variation if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106056.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106057.png" /> is quasi-additive and of bounded variation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106058.png" />, then both functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106060.png" /> are quasi-additive on all subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106061.png" />.
+
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]]].
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106062.png" /> coincides with the Burkill–Cesari measure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b111/b111060/b11106063.png" />.
+
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]]).
 
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]]).
Line 53: Line 124:
  
 
====References====
 
====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>
+
<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 $\int f(\xi,\phi)$"  ''Ann. Scuola Norm. Sup. Pisa'' , '''22'''  (1968)  pp. 163–191</TD></TR></table>

Latest revision as of 08:49, 26 March 2023


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 $\int f(\xi,\phi)$" Ann. Scuola Norm. Sup. Pisa , 22 (1968) pp. 163–191
How to Cite This Entry:
Burkill-Cesari integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Burkill-Cesari_integral&oldid=22219
This article was adapted from an original article by P. Brandi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article