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''on a group''
 
''on a group''
  
Integration of functions on a [[Topological group|topological group]] that has a certain invariant property with respect to the group operations. Thus, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i0522401.png" /> be a locally compact topological group, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i0522402.png" /> be the vector space of all continuous complex-valued functions with compact support on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i0522403.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i0522404.png" /> be an integral on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i0522405.png" />, that is, a positive [[Linear functional|linear functional]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i0522406.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i0522407.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i0522408.png" />). The integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i0522409.png" /> is called left-invariant (or right-invariant) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224010.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224011.png" />) for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224013.png" />; here
+
Integration of functions on a [[Topological group|topological group]] that has a certain invariant property with respect to the group operations. Thus, let $  G $
 +
be a locally compact topological group, let $  C _ {0} ( G) $
 +
be the vector space of all continuous complex-valued functions with compact support on $  G $
 +
and let $  I $
 +
be an integral on $  C _ {0} ( G) $,  
 +
that is, a positive [[Linear functional|linear functional]] on $  C _ {0} ( G) $(
 +
$  I f \geq  0 $
 +
for $  f \geq  0 $).  
 +
The integral $  I $
 +
is called left-invariant (or right-invariant) if $  I ( gf  ) = If $(
 +
or $  I ( fg ) = If  $)  
 +
for all $  g \in G $,  
 +
$  f \in C _ {0} ( G) $;  
 +
here
  
<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/i/i052/i052240/i05224014.png" /></td> </tr></table>
+
$$
 +
( gf  ) ( x )  = f ( g  ^ {-} 1 x ) ,\ \
 +
( fg ) ( x)  = f ( x g ) .
 +
$$
  
The integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224015.png" /> is called (two-sided) invariant if it is both left- and right-invariant. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224018.png" />, defines a one-to-one correspondence between the classes of left- and right-invariant integrals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224019.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224021.png" /> is called inversion invariant.
+
The integral $  I $
 +
is called (two-sided) invariant if it is both left- and right-invariant. The mapping $  I \rightarrow \widehat{I}  $,  
 +
where $  \widehat{I}  f = I \widehat{f}  $,  
 +
$  \widehat{f}  ( x) = f ( x  ^ {-} 1 ) $,  
 +
defines a one-to-one correspondence between the classes of left- and right-invariant integrals on $  C _ {0} ( G) $.  
 +
If $  I = \widehat{I}  $,  
 +
then $  I $
 +
is called inversion invariant.
  
There exists on every locally compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224022.png" /> a non-zero left-invariant integral; it is unique up to a numerical factor (the Haar–von Neumann–Weil theorem). This integral is called the left Haar integral. The following equation holds:
+
There exists on every locally compact group $  G $
 +
a non-zero left-invariant integral; it is unique up to a numerical factor (the Haar–von Neumann–Weil theorem). This integral is called the left Haar integral. The following equation holds:
  
<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/i/i052/i052240/i05224023.png" /></td> </tr></table>
+
$$
 +
I ( fg )  = \Delta ( g ) I f ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224026.png" /> is a continuous homomorphism from the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224027.png" /> into the multiplicative group of positive real numbers (a positive character). Furthermore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224028.png" />. The character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224029.png" /> is called the modulus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224030.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224031.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224032.png" /> is called unimodular. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224033.png" /> is a two-sided invariant integral.
+
where $  g \in G $,  
 +
$  f \in C _ {0} ( G ) $
 +
and $  \Delta $
 +
is a continuous homomorphism from the group $  G $
 +
into the multiplicative group of positive real numbers (a positive character). Furthermore, $  \widehat{I}  f = I ( f / \Delta ) $.  
 +
The character $  \Delta $
 +
is called the modulus of $  G $.  
 +
If $  \Delta ( g) \equiv 1 $,  
 +
then $  G $
 +
is called unimodular. In this case $  I $
 +
is a two-sided invariant integral.
  
In particular, every compact group (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224035.png" />) and every discrete group (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224037.png" />) is unimodular.
+
In particular, every compact group (where $  I1 < \infty $,  
 +
$  \widehat{I}  = I $)  
 +
and every discrete group (where $  If = \sum _ {g} f ( g) $,  
 +
$  f \in C _ {0} ( G) $)  
 +
is unimodular.
  
According to the Riesz theorem, every integral on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224038.png" /> is a [[Lebesgue integral|Lebesgue integral]] with respect to some [[Borel measure|Borel measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224039.png" /> which is uniquely defined in the class of Borel measures that are finite on each compact subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224040.png" />. The left- (or right-) invariant measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224041.png" /> corresponding to the left (right) Haar integral on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224042.png" /> is called the left (right) [[Haar measure|Haar measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224043.png" />.
+
According to the Riesz theorem, every integral on $  C _ {0} ( G) $
 +
is a [[Lebesgue integral|Lebesgue integral]] with respect to some [[Borel measure|Borel measure]] $  \mu $
 +
which is uniquely defined in the class of Borel measures that are finite on each compact subset $  K \subset  G $.  
 +
The left- (or right-) invariant measure $  \mu $
 +
corresponding to the left (right) Haar integral on $  C _ {0} ( G) $
 +
is called the left (right) [[Haar measure|Haar measure]] on $  G $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224044.png" /> be a closed subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224045.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224046.png" /> be the modulus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224047.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224048.png" /> can be extended to a continuous positive character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224049.png" /> (cf. [[Character of a group|Character of a group]]), then there exists on the left homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224050.png" /> a relatively invariant integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224051.png" />, that is, a positive functional on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224052.png" /> of continuous functions with compact support on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224053.png" /> that satisfies the identity
+
Let $  H $
 +
be a closed subgroup of $  G $
 +
and let $  \Delta _ {0} $
 +
be the modulus of $  H $.  
 +
If $  \Delta _ {0} $
 +
can be extended to a continuous positive character of $  G $(
 +
cf. [[Character of a group|Character of a group]]), then there exists on the left homogeneous space $  X = G / H $
 +
a relatively invariant integral $  J $,  
 +
that is, a positive functional on the space $  C _ {0} ( X) $
 +
of continuous functions with compact support on $  X $
 +
that satisfies the identity
  
<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/i/i052/i052240/i05224054.png" /></td> </tr></table>
+
$$
 +
J ( gf  )  = \delta ( g ) J f
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224056.png" />; here
+
for all $  g \in G $,  
 +
$  f \in C _ {0} ( X) $;  
 +
here
  
<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/i/i052/i052240/i05224057.png" /></td> </tr></table>
+
$$
 +
( gf  ) ( x)  = f ( g  ^ {-} 1 x ) ,\ \
 +
\delta ( g)  =
 +
\frac{\Delta _ {0} ( g) }{\Delta ( g) }
 +
,
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224058.png" /> is the modulus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224059.png" />. This integral is defined by the rule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224060.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224061.png" /> is the left Haar integral on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224063.png" /> is a function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224064.png" /> such that
+
and $  \Delta $
 +
is the modulus of $  G $.  
 +
This integral is defined by the rule $  J f = I ( \delta \widetilde{f}  ) $,  
 +
where $  I $
 +
is the left Haar integral on $  G $
 +
and $  \widetilde{f}  $
 +
is a function on $  G $
 +
such that
  
<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/i/i052/i052240/i05224065.png" /></td> </tr></table>
+
$$
 +
f ( gH )  = I _ {0} (( g \widetilde{f}  ) _ {H} ) .
 +
$$
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224066.png" /> is the left Haar integral on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224068.png" /> is the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224069.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224070.png" />.) This is well-defined since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224071.png" /> is a mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224072.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224074.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052240/i05224075.png" />. The notion of an invariant mean (cf. [[Invariant average|Invariant average]]) is closely related to that of invariant integration.
+
( $  I _ {0} $
 +
is the left Haar integral on $  H $
 +
and $  \phi _ {H} $
 +
is the restriction of $  \phi $
 +
to $  H $.)  
 +
This is well-defined since $  \widetilde{f}  \rightarrow f $
 +
is a mapping from $  C _ {0} ( G) $
 +
onto $  C _ {0} ( X) $
 +
and $  Jf = 0 $
 +
when $  f = 0 $.  
 +
The notion of an invariant mean (cf. [[Invariant average|Invariant average]]) is closely related to that of invariant integration.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Integration" , Addison-Wesley  (1975)  pp. Chapt.6;7;8  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Weil,  "l'Intégration dans les groupes topologiques et ses applications" , Hermann  (1940)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.H. Loomis,  "An introduction to abstract harmonic analysis" , v. Nostrand  (1953)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E. Hewitt,  K.A. Ross,  "Abstract harmonic analysis" , '''1''' , Springer  (1979)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Integration" , Addison-Wesley  (1975)  pp. Chapt.6;7;8  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Weil,  "l'Intégration dans les groupes topologiques et ses applications" , Hermann  (1940)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.H. Loomis,  "An introduction to abstract harmonic analysis" , v. Nostrand  (1953)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E. Hewitt,  K.A. Ross,  "Abstract harmonic analysis" , '''1''' , Springer  (1979)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Reiter,  "Classical harmonic analysis and locally compact groups" , Clarendon Press  (1968)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Reiter,  "Classical harmonic analysis and locally compact groups" , Clarendon Press  (1968)</TD></TR></table>

Latest revision as of 22:13, 5 June 2020


on a group

Integration of functions on a topological group that has a certain invariant property with respect to the group operations. Thus, let $ G $ be a locally compact topological group, let $ C _ {0} ( G) $ be the vector space of all continuous complex-valued functions with compact support on $ G $ and let $ I $ be an integral on $ C _ {0} ( G) $, that is, a positive linear functional on $ C _ {0} ( G) $( $ I f \geq 0 $ for $ f \geq 0 $). The integral $ I $ is called left-invariant (or right-invariant) if $ I ( gf ) = If $( or $ I ( fg ) = If $) for all $ g \in G $, $ f \in C _ {0} ( G) $; here

$$ ( gf ) ( x ) = f ( g ^ {-} 1 x ) ,\ \ ( fg ) ( x) = f ( x g ) . $$

The integral $ I $ is called (two-sided) invariant if it is both left- and right-invariant. The mapping $ I \rightarrow \widehat{I} $, where $ \widehat{I} f = I \widehat{f} $, $ \widehat{f} ( x) = f ( x ^ {-} 1 ) $, defines a one-to-one correspondence between the classes of left- and right-invariant integrals on $ C _ {0} ( G) $. If $ I = \widehat{I} $, then $ I $ is called inversion invariant.

There exists on every locally compact group $ G $ a non-zero left-invariant integral; it is unique up to a numerical factor (the Haar–von Neumann–Weil theorem). This integral is called the left Haar integral. The following equation holds:

$$ I ( fg ) = \Delta ( g ) I f , $$

where $ g \in G $, $ f \in C _ {0} ( G ) $ and $ \Delta $ is a continuous homomorphism from the group $ G $ into the multiplicative group of positive real numbers (a positive character). Furthermore, $ \widehat{I} f = I ( f / \Delta ) $. The character $ \Delta $ is called the modulus of $ G $. If $ \Delta ( g) \equiv 1 $, then $ G $ is called unimodular. In this case $ I $ is a two-sided invariant integral.

In particular, every compact group (where $ I1 < \infty $, $ \widehat{I} = I $) and every discrete group (where $ If = \sum _ {g} f ( g) $, $ f \in C _ {0} ( G) $) is unimodular.

According to the Riesz theorem, every integral on $ C _ {0} ( G) $ is a Lebesgue integral with respect to some Borel measure $ \mu $ which is uniquely defined in the class of Borel measures that are finite on each compact subset $ K \subset G $. The left- (or right-) invariant measure $ \mu $ corresponding to the left (right) Haar integral on $ C _ {0} ( G) $ is called the left (right) Haar measure on $ G $.

Let $ H $ be a closed subgroup of $ G $ and let $ \Delta _ {0} $ be the modulus of $ H $. If $ \Delta _ {0} $ can be extended to a continuous positive character of $ G $( cf. Character of a group), then there exists on the left homogeneous space $ X = G / H $ a relatively invariant integral $ J $, that is, a positive functional on the space $ C _ {0} ( X) $ of continuous functions with compact support on $ X $ that satisfies the identity

$$ J ( gf ) = \delta ( g ) J f $$

for all $ g \in G $, $ f \in C _ {0} ( X) $; here

$$ ( gf ) ( x) = f ( g ^ {-} 1 x ) ,\ \ \delta ( g) = \frac{\Delta _ {0} ( g) }{\Delta ( g) } , $$

and $ \Delta $ is the modulus of $ G $. This integral is defined by the rule $ J f = I ( \delta \widetilde{f} ) $, where $ I $ is the left Haar integral on $ G $ and $ \widetilde{f} $ is a function on $ G $ such that

$$ f ( gH ) = I _ {0} (( g \widetilde{f} ) _ {H} ) . $$

( $ I _ {0} $ is the left Haar integral on $ H $ and $ \phi _ {H} $ is the restriction of $ \phi $ to $ H $.) This is well-defined since $ \widetilde{f} \rightarrow f $ is a mapping from $ C _ {0} ( G) $ onto $ C _ {0} ( X) $ and $ Jf = 0 $ when $ f = 0 $. The notion of an invariant mean (cf. Invariant average) is closely related to that of invariant integration.

References

[1] N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French)
[2] A. Weil, "l'Intégration dans les groupes topologiques et ses applications" , Hermann (1940)
[3] L.H. Loomis, "An introduction to abstract harmonic analysis" , v. Nostrand (1953)
[4] E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1 , Springer (1979)

Comments

References

[a1] H. Reiter, "Classical harmonic analysis and locally compact groups" , Clarendon Press (1968)
How to Cite This Entry:
Invariant integration. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Invariant_integration&oldid=16515
This article was adapted from an original article by D.P. Zhelobenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article