Namespaces
Variants
Actions

Difference between revisions of "Daniell integral"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (fix formula)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
 +
<!--
 +
d0301101.png
 +
$#A+1 = 68 n = 0
 +
$#C+1 = 68 : ~/encyclopedia/old_files/data/D030/D.0300110 Daniell integral
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
An extension of the concept of the integral, proposed by P. Daniell [[#References|[1]]]. The construction scheme of this integral, which is known as the Daniell scheme, is an extension of the integral originally defined for a certain set of functions — the so-called elementary functions — to a wider class of functions. While retaining the way of extending but by changing the content of the initial set of elementary functions, it is possible to arrive at different extensions of the concept of the integral. In this scheme the concept of an elementary integral is axiomatically defined, unlike in Lebesgue's scheme (cf. [[Lebesgue integral|Lebesgue integral]]) in which the concept of a measure is axiomatic.
 
An extension of the concept of the integral, proposed by P. Daniell [[#References|[1]]]. The construction scheme of this integral, which is known as the Daniell scheme, is an extension of the integral originally defined for a certain set of functions — the so-called elementary functions — to a wider class of functions. While retaining the way of extending but by changing the content of the initial set of elementary functions, it is possible to arrive at different extensions of the concept of the integral. In this scheme the concept of an elementary integral is axiomatically defined, unlike in Lebesgue's scheme (cf. [[Lebesgue integral|Lebesgue integral]]) in which the concept of a measure is axiomatic.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d0301101.png" /> be an arbitrary set and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d0301102.png" /> be a certain set of real bounded functions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d0301103.png" />; these functions are called elementary. It is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d0301104.png" /> is a vector lattice, i.e. from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d0301105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d0301106.png" /> it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d0301107.png" />, and
+
Let $  X $
 +
be an arbitrary set and let $  L _ {0} $
 +
be a certain set of real bounded functions defined on $  X $;  
 +
these functions are called elementary. It is assumed that $  L _ {0} $
 +
is a vector lattice, i.e. from $  f, g \in L _ {0} $
 +
and $  \alpha , \beta \in \mathbf R $
 +
it follows that $  \alpha f+ \beta g \in L _ {0} $,  
 +
and
  
<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/d/d030/d030110/d0301108.png" /></td> </tr></table>
+
$$
 +
f, g  \in  L _ {0} \
 +
\textrm{ implies } \
 +
\sup ( f, g) , \inf ( f, g)  \in  L _ {0} .
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d0301109.png" /> be a real functional defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011010.png" /> such that
+
Let $  I $
 +
be a real functional defined on $  L _ {0} $
 +
such that
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011011.png" /> (linearity);
+
1) $  I( \alpha f + \beta g) = \alpha I ( f  ) + \beta I ( g) $(
 +
linearity);
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011012.png" /> (non-negativity);
+
2) $  f \geq  0 \Rightarrow  I( f  ) \geq  0 $(
 +
non-negativity);
  
3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011013.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011014.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011015.png" /> (continuity with respect to monotone convergence).
+
3) if $  f _ {n} ( x) \downarrow 0 $
 +
for all $  x $,  
 +
then $  I ( f _ {n} ) \rightarrow 0 $(
 +
continuity with respect to monotone convergence).
  
Such a functional is known as an integral over elementary functions or an elementary integral. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011016.png" /> is said to be a set of measure zero if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011017.png" /> there exists a non-decreasing sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011018.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011019.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011020.png" />, and
+
Such a functional is known as an integral over elementary functions or an elementary integral. A set $  M \subset  X $
 +
is said to be a set of measure zero if for each $  \epsilon > 0 $
 +
there exists a non-decreasing sequence $  \{ g _ {n} \} \subset  L _ {0} $
 +
such that $  \sup _ {n}  g _ {n} ( x) \geq  \chi _ {M} ( x) $
 +
for all $  x $,  
 +
and
  
<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/d/d030/d030110/d03011021.png" /></td> </tr></table>
+
$$
 +
\sup  I ( g _ {n} )  \leq  \epsilon .
 +
$$
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011022.png" /> denotes the characteristic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011023.png" />.
+
Here, $  \chi _ {M} $
 +
denotes the characteristic function of $  M $.
  
A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011024.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011025.png" /> belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011027.png" /> if there exists a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011028.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011029.png" /> [[Almost-everywhere|almost-everywhere]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011030.png" />. The number
+
A function $  f $
 +
defined on $  X $
 +
belongs to the class $  L  ^ {+} $
 +
if there exists a sequence $  \{ f _ {n} \} \in L _ {0} $
 +
such that $  f _ {n} ( x) \uparrow f( x) $[[
 +
Almost-everywhere|almost-everywhere]] and $  I ( f _ {n} ) \leq  c < + \infty $.  
 +
The number
  
<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/d/d030/d030110/d03011031.png" /></td> </tr></table>
+
$$
 +
I( f  )  = \lim\limits _ { n }  I( f _ {n} )
 +
$$
  
is said to be the integral of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011033.png" />. The integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011034.png" /> does not depend on the choice of the particular approximating sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011035.png" />.
+
is said to be the integral of $  f $.  
 +
The integral $  I( f  ) $
 +
does not depend on the choice of the particular approximating sequence $  \{ f _ {n} \} $.
  
The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011037.png" /> is the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011038.png" /> which are defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011039.png" /> and which are representable in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011040.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011041.png" />. Functions of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011042.png" /> are called summable, while the number
+
The class $  L $
 +
is the set of functions $  f $
 +
which are defined on $  X $
 +
and which are representable in the form $  f= f _ {1} - f _ {2} $,  
 +
where $  f _ {1} , f _ {2} \in L  ^ {+} $.  
 +
Functions of the class $  L $
 +
are called summable, while the number
  
<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/d/d030/d030110/d03011043.png" /></td> </tr></table>
+
$$
 +
I( f  )  = I( f _ {1} ) - I( f _ {2} )
 +
$$
  
is known as the Daniell integral of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011045.png" />. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011046.png" /> is a vector lattice of finite functions (considered up to sets of measure zero) which is closed with respect to almost-everywhere convergence, with finite integrals, while the Daniell integral of summable functions has the properties of linearity, non-negativity, continuity with respect to almost-everywhere convergence of majorable summable functions (Lebesgue's theorem on passing to the limit under the integral sign), and also several other natural properties of the integral.
+
is known as the Daniell integral of the function $  f $.  
 +
The class $  L $
 +
is a vector lattice of finite functions (considered up to sets of measure zero) which is closed with respect to almost-everywhere convergence, with finite integrals, while the Daniell integral of summable functions has the properties of linearity, non-negativity, continuity with respect to almost-everywhere convergence of majorable summable functions (Lebesgue's theorem on passing to the limit under the integral sign), and also several other natural properties of the integral.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011048.png" /> is the set of the step functions
+
If $  X= [ a, b] $
 +
and $  L _ {0} $
 +
is the set of the step functions
  
<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/d/d030/d030110/d03011049.png" /></td> </tr></table>
+
$$
 +
f( x)  = c _ {i} , a _ {i} \leq  x < b _ {i} ,\  \cup _ { i= 1 }^ { n }  [ a _ {i} , b _ {i} )  = [ a, b), b _ {i} = a _ {i+ 1} ,
 +
$$
  
the Daniell integral becomes identical with the Lebesgue integral on the summable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011050.png" />. The Daniell scheme may be used to construct the integral of functions with values in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011051.png" />-complete lattice.
+
the Daniell integral becomes identical with the Lebesgue integral on the summable functions on $  [ a, b] $.  
 +
The Daniell scheme may be used to construct the integral of functions with values in a $  \sigma $-
 +
complete lattice.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Daniell,  "A general form of integral"  ''Ann. of Math'' , '''19'''  (1917)  pp. 279–294</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.E. Shilov,  B.L. Gurevich,  "Integral, measure, and derivative: a unified approach" , Dover, reprint  (1977)  (Translated from Russian)</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></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Daniell,  "A general form of integral"  ''Ann. of Math'' , '''19'''  (1917)  pp. 279–294</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.E. Shilov,  B.L. Gurevich,  "Integral, measure, and derivative: a unified approach" , Dover, reprint  (1977)  (Translated from Russian)</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></table>
 
 
  
 
====Comments====
 
====Comments====
Property 3) to be satisfied by the non-negative linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011052.png" /> above (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011053.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011054.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011055.png" />) is called the Denjoy condition, and is a very important requirement.
+
Property 3) to be satisfied by the non-negative linear functional $  I $
 +
above (i.e. $  I ( f _ {n} ) \rightarrow 0 $
 +
as $  f _ {n} ( x) \downarrow 0 $
 +
for all $  x $)  
 +
is called the Denjoy condition, and is a very important requirement.
  
In the article above, functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011056.png" /> (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011057.png" />) differing on a set of measure zero are tacitly identified; the equivalence classes thus obtained are also called functions (with some abuse of language), as is usually done in measure theory. The statement that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011058.png" /> be a vector lattice is thus to be understood as: the set of equivalence classes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011059.png" /> forms a vector lattice.
+
In the article above, functions in $  L  ^ {+} $(
 +
in $  L $)  
 +
differing on a set of measure zero are tacitly identified; the equivalence classes thus obtained are also called functions (with some abuse of language), as is usually done in measure theory. The statement that $  L $
 +
be a vector lattice is thus to be understood as: the set of equivalence classes in $  L $
 +
forms a vector lattice.
  
If the [[Vector lattice|vector lattice]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011060.png" /> has the property
+
If the [[Vector lattice|vector lattice]] $  L _ {0} $
 +
has the property
  
<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/d/d030/d030110/d03011061.png" /></td> </tr></table>
+
$$
 +
f \in L _ {0}  \textrm{ implies }  \inf ( 1, f  ) \in L _ {0} ,
 +
$$
  
then there is on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011062.png" />-field generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011063.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011064.png" /> a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011065.png" />-finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011066.png" />-additive [[Measure|measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011067.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011068.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011069.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011070.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011071.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030110/d03011072.png" /> (see [[#References|[3]]]). Actually, the Daniell integral is often used for constructing measures in functional analysis.
+
then there is on the $  \sigma $-
 +
field generated by $  L _ {0} $
 +
on $  X $
 +
a unique $  \sigma $-
 +
finite $  \sigma $-
 +
additive [[Measure|measure]] $  \mu $
 +
such that $  L $
 +
is $  L _ {1} ( \mu ) $,  
 +
and $  I ( f  ) $
 +
is $  \int f  d \mu $
 +
for $  f \in L $(
 +
see [[#References|[3]]]). Actually, the Daniell integral is often used for constructing measures in functional analysis.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981)  pp. 199–206; 334</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Szökefalvi-Nagy,  "Real functions and orthogonal expansions" , Oxford Univ. Press  (1965)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981)  pp. 199–206; 334</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Szökefalvi-Nagy,  "Real functions and orthogonal expansions" , Oxford Univ. Press  (1965)</TD></TR></table>

Latest revision as of 13:31, 26 March 2023


An extension of the concept of the integral, proposed by P. Daniell [1]. The construction scheme of this integral, which is known as the Daniell scheme, is an extension of the integral originally defined for a certain set of functions — the so-called elementary functions — to a wider class of functions. While retaining the way of extending but by changing the content of the initial set of elementary functions, it is possible to arrive at different extensions of the concept of the integral. In this scheme the concept of an elementary integral is axiomatically defined, unlike in Lebesgue's scheme (cf. Lebesgue integral) in which the concept of a measure is axiomatic.

Let $ X $ be an arbitrary set and let $ L _ {0} $ be a certain set of real bounded functions defined on $ X $; these functions are called elementary. It is assumed that $ L _ {0} $ is a vector lattice, i.e. from $ f, g \in L _ {0} $ and $ \alpha , \beta \in \mathbf R $ it follows that $ \alpha f+ \beta g \in L _ {0} $, and

$$ f, g \in L _ {0} \ \textrm{ implies } \ \sup ( f, g) , \inf ( f, g) \in L _ {0} . $$

Let $ I $ be a real functional defined on $ L _ {0} $ such that

1) $ I( \alpha f + \beta g) = \alpha I ( f ) + \beta I ( g) $( linearity);

2) $ f \geq 0 \Rightarrow I( f ) \geq 0 $( non-negativity);

3) if $ f _ {n} ( x) \downarrow 0 $ for all $ x $, then $ I ( f _ {n} ) \rightarrow 0 $( continuity with respect to monotone convergence).

Such a functional is known as an integral over elementary functions or an elementary integral. A set $ M \subset X $ is said to be a set of measure zero if for each $ \epsilon > 0 $ there exists a non-decreasing sequence $ \{ g _ {n} \} \subset L _ {0} $ such that $ \sup _ {n} g _ {n} ( x) \geq \chi _ {M} ( x) $ for all $ x $, and

$$ \sup I ( g _ {n} ) \leq \epsilon . $$

Here, $ \chi _ {M} $ denotes the characteristic function of $ M $.

A function $ f $ defined on $ X $ belongs to the class $ L ^ {+} $ if there exists a sequence $ \{ f _ {n} \} \in L _ {0} $ such that $ f _ {n} ( x) \uparrow f( x) $[[ Almost-everywhere|almost-everywhere]] and $ I ( f _ {n} ) \leq c < + \infty $. The number

$$ I( f ) = \lim\limits _ { n } I( f _ {n} ) $$

is said to be the integral of $ f $. The integral $ I( f ) $ does not depend on the choice of the particular approximating sequence $ \{ f _ {n} \} $.

The class $ L $ is the set of functions $ f $ which are defined on $ X $ and which are representable in the form $ f= f _ {1} - f _ {2} $, where $ f _ {1} , f _ {2} \in L ^ {+} $. Functions of the class $ L $ are called summable, while the number

$$ I( f ) = I( f _ {1} ) - I( f _ {2} ) $$

is known as the Daniell integral of the function $ f $. The class $ L $ is a vector lattice of finite functions (considered up to sets of measure zero) which is closed with respect to almost-everywhere convergence, with finite integrals, while the Daniell integral of summable functions has the properties of linearity, non-negativity, continuity with respect to almost-everywhere convergence of majorable summable functions (Lebesgue's theorem on passing to the limit under the integral sign), and also several other natural properties of the integral.

If $ X= [ a, b] $ and $ L _ {0} $ is the set of the step functions

$$ f( x) = c _ {i} , a _ {i} \leq x < b _ {i} ,\ \cup _ { i= 1 }^ { n } [ a _ {i} , b _ {i} ) = [ a, b), b _ {i} = a _ {i+ 1} , $$

the Daniell integral becomes identical with the Lebesgue integral on the summable functions on $ [ a, b] $. The Daniell scheme may be used to construct the integral of functions with values in a $ \sigma $- complete lattice.

References

[1] P. Daniell, "A general form of integral" Ann. of Math , 19 (1917) pp. 279–294
[2] G.E. Shilov, B.L. Gurevich, "Integral, measure, and derivative: a unified approach" , Dover, reprint (1977) (Translated from Russian)
[3] L.H. Loomis, "An introduction to abstract harmonic analysis" , v. Nostrand (1953)

Comments

Property 3) to be satisfied by the non-negative linear functional $ I $ above (i.e. $ I ( f _ {n} ) \rightarrow 0 $ as $ f _ {n} ( x) \downarrow 0 $ for all $ x $) is called the Denjoy condition, and is a very important requirement.

In the article above, functions in $ L ^ {+} $( in $ L $) differing on a set of measure zero are tacitly identified; the equivalence classes thus obtained are also called functions (with some abuse of language), as is usually done in measure theory. The statement that $ L $ be a vector lattice is thus to be understood as: the set of equivalence classes in $ L $ forms a vector lattice.

If the vector lattice $ L _ {0} $ has the property

$$ f \in L _ {0} \textrm{ implies } \inf ( 1, f ) \in L _ {0} , $$

then there is on the $ \sigma $- field generated by $ L _ {0} $ on $ X $ a unique $ \sigma $- finite $ \sigma $- additive measure $ \mu $ such that $ L $ is $ L _ {1} ( \mu ) $, and $ I ( f ) $ is $ \int f d \mu $ for $ f \in L $( see [3]). Actually, the Daniell integral is often used for constructing measures in functional analysis.

References

[a1] K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981) pp. 199–206; 334
[a2] B. Szökefalvi-Nagy, "Real functions and orthogonal expansions" , Oxford Univ. Press (1965)
How to Cite This Entry:
Daniell integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Daniell_integral&oldid=17172
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article