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A finitely-additive [[Set function|set function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v0964901.png" /> defined on a field of subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v0964902.png" /> of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v0964903.png" />, with values in a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v0964904.png" /> (or, more generally, a topological vector space). A vector measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v0964905.png" /> is called strongly additive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v0964906.png" /> converges in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v0964907.png" /> for every sequence of pairwise disjoint sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v0964908.png" />, and countably additive if, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v0964909.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649010.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649011.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649012.png" /> is countably additive for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649013.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649014.png" /> is said to be weakly countably additive. A weakly countably-additive vector measure defined on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649015.png" />-field is countably additive (the Orlicz–Pettis theorem). The variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649017.png" /> is the extended real-valued non-negative finitely-additive set function defined by
+
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<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/v/v096/v096490/v09649018.png" /></td> </tr></table>
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{{TEX|auto}}
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{{TEX|done}}
  
where the supremum is over all finite partitions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649019.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649020.png" /> into disjoint members of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649021.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649022.png" /> is said to have bounded variation if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649023.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649024.png" /> is countably additive if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649025.png" /> is. The semi-variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649026.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649027.png" /> is defined by
+
A finitely-additive [[Set function|set function]]  $  F $
 +
defined on a field of subsets  $  {\mathcal F} $
 +
of a set  $  \Omega $,
 +
with values in a [[Banach space|Banach space]]  $  X $(
 +
or, more generally, a topological vector space). A vector measure  $  F $
 +
is called strongly additive if  $  \sum _ {n=1} ^  \infty  F( E _ {n} ) $
 +
converges in  $  X $
 +
for every sequence of pairwise disjoint sets  $  E _ {n} \in {\mathcal F} $,
 +
and countably additive if, in addition,  $  \sum _ {n=1}  ^  \infty  F( E _ {n} ) = F ( \cup _ {n=1}  ^  \infty  E _ {n} ) $
 +
whenever  $  \cup _ {n=1}  ^  \infty  E _ {n} $
 +
belongs to  $  {\mathcal F} $.  
 +
If  $  x  ^ {*} F $
 +
is countably additive for every  $  x  ^ {*} \in X  ^ {*} $,
 +
then  $  F $
 +
is said to be weakly countably additive. A weakly countably-additive vector measure defined on a  $  \sigma $-
 +
field is countably additive (the Orlicz–Pettis theorem). The variation $  | F | $
 +
of $  F $
 +
is the extended real-valued non-negative finitely-additive set function defined by
  
<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/v/v096/v096490/v09649028.png" /></td> </tr></table>
+
$$
 +
| F | ( E)  = \sup _  \pi  \
 +
\sum _ {A \in \pi } \| F( A) \| ,\ \
 +
E \in {\mathcal F} ,
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649029.png" /> is a monotone finitely-subadditive set function, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649030.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649031.png" /> is said to have bounded semi-variation. Since, equivalently, this means that the range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649032.png" /> is norm bounded, such measures are also called bounded. Vector measures of bounded variation are strongly additive, and strongly-additive vector measures are bounded. A bounded vector measure is strongly additive if and only if its range is relatively weakly compact. In particular, a countably-additive vector measure has relatively weakly-compact range.
+
where the supremum is over all finite partitions  $  \pi $
 +
of  $  E $
 +
into disjoint members of  $  {\mathcal F} $.  
 +
$  F $
 +
is said to have bounded variation if  $  | F | ( \Omega ) < \infty $.  
 +
$  | F | $
 +
is countably additive if and only if $  F $
 +
is. The semi-variation  $  \| F \| $
 +
of  $  F $
 +
is defined by
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649033.png" /> be a sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649034.png" />-valued countably-additive vector measures defined on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649035.png" />-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649036.png" />, and let each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649037.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649038.png" />-continuous, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649039.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649040.png" /> is a non-negative extended real-valued measure. Now, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649041.png" /> exists for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649042.png" />, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649043.png" />-continuity is uniform for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649044.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649045.png" />, uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649046.png" />. Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649047.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649048.png" />-continuous. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649049.png" /> is finite it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649050.png" /> is countably additive. This is the Vitali–Hahn–Saks theorem. Another striking result from the theory of vector measures is the so-called Nikodým boundedness theorem: For a collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649051.png" /> of bounded vector measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649052.png" /> on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649053.png" />-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649054.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649055.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649056.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649057.png" /> is uniformly bounded, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649058.png" />. There are also versions for strongly-additive vector measures of the well-known decomposition theorems of Yosida–Hewitt and of Lebesgue (see [[#References|[a3]]]). Finally, a non-atomic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649059.png" />-valued measure on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649060.png" />-field has compact and convex range if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649061.png" />. This is Lyapunov's theorem. It fails for infinite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649062.png" />.
+
$$
 +
\| F \| ( E)  = \sup \{ {| x  ^ {*} F | ( E) } : {
 +
\| x  ^ {*} \| \leq  1 } \}
 +
,\ \
 +
E \in {\mathcal F} .
 +
$$
 +
 
 +
$  \| F \| $
 +
is a monotone finitely-subadditive set function, and if  $  \| F \| ( \Omega ) < \infty $,
 +
then  $  F $
 +
is said to have bounded semi-variation. Since, equivalently, this means that the range of  $  F $
 +
is norm bounded, such measures are also called bounded. Vector measures of bounded variation are strongly additive, and strongly-additive vector measures are bounded. A bounded vector measure is strongly additive if and only if its range is relatively weakly compact. In particular, a countably-additive vector measure has relatively weakly-compact range.
 +
 
 +
Let  $  ( F _ {n} ) $
 +
be a sequence of $  X $-
 +
valued countably-additive vector measures defined on a $  \sigma $-
 +
field $  \Sigma $,  
 +
and let each $  F _ {n} $
 +
be $  \mu $-
 +
continuous, i.e. $  \lim\limits _ {\mu ( E) \rightarrow 0 }  F _ {n} ( E) = 0 $,  
 +
where $  \mu $
 +
is a non-negative extended real-valued measure. Now, if $  \lim\limits _ {n \rightarrow \infty }  F _ {n} ( E) = F( E) $
 +
exists for every $  E \in \Sigma $,  
 +
then the $  \mu $-
 +
continuity is uniform for $  n \in \mathbf N $,  
 +
i.e. $  \lim\limits _ {\mu ( E) \rightarrow 0 }  F _ {n} ( E) = 0 $,  
 +
uniformly in $  n $.  
 +
Hence $  F $
 +
is $  \mu $-
 +
continuous. In particular, if $  \mu $
 +
is finite it follows that $  F $
 +
is countably additive. This is the Vitali–Hahn–Saks theorem. Another striking result from the theory of vector measures is the so-called Nikodým boundedness theorem: For a collection $  M $
 +
of bounded vector measures $  F $
 +
on a $  \sigma $-
 +
field $  \Sigma $,  
 +
if $  \sup _ {F \in M }  \| F( E) \| < \infty $
 +
for each $  E \in \Sigma $,  
 +
then $  M $
 +
is uniformly bounded, i.e. $  \sup _ {F \in M , E \in \Sigma }  \| F( E) \| < \infty $.  
 +
There are also versions for strongly-additive vector measures of the well-known decomposition theorems of Yosida–Hewitt and of Lebesgue (see [[#References|[a3]]]). Finally, a non-atomic $  X $-
 +
valued measure on a $  \sigma $-
 +
field has compact and convex range if $  \mathop{\rm dim}  X < \infty $.  
 +
This is Lyapunov's theorem. It fails for infinite-dimensional $  X $.
  
 
Vector measure theory has important applications to other areas of functional analysis. First of all to operator theory, where problems of representing operators on certain function spaces may well have been the original motive for studying vector measures. Much later, in the 1970s, the problem of differentiating vector measures led to a body of results in the geometry of Banach spaces, centering around the so-called Radon–Nikodým property. Below these developments are given briefly (see also [[#References|[a1]]] and [[#References|[a4]]]); see [[#References|[a5]]] for the role of vector measures in control theory.
 
Vector measure theory has important applications to other areas of functional analysis. First of all to operator theory, where problems of representing operators on certain function spaces may well have been the original motive for studying vector measures. Much later, in the 1970s, the problem of differentiating vector measures led to a body of results in the geometry of Banach spaces, centering around the so-called Radon–Nikodým property. Below these developments are given briefly (see also [[#References|[a1]]] and [[#References|[a4]]]); see [[#References|[a5]]] for the role of vector measures in control theory.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649063.png" /> be a compact Hausdorff space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649064.png" /> the space of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649065.png" /> with the sup-norm, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649066.png" /> a bounded linear operator (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649067.png" /> is any Banach space). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649068.png" /> can be represented by a weak-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649069.png" /> countably-additive vector measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649070.png" /> defined on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649071.png" />-field of Borel sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649072.png" /> and taking its values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649073.png" />, the bidual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649074.png" /> (cf. [[Adjoint space|Adjoint space]]). This representation is particularly satisfactory when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649075.png" /> is weakly compact, for then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649076.png" /> has its values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649077.png" />, and is countably additive (either of these properties is in fact equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649078.png" /> being weakly compact). Then one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649079.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649080.png" />), where the integral has its more or less obvious meaning. An immediate consequence of this representation formula is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649081.png" /> maps weakly-compact sets into norm-compact sets (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649082.png" /> has the Dunford–Pettis property). Other classes of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649083.png" /> such as the compact, the nuclear and the absolutely summing ones admit equally nice characterizations in terms of their representing measures (see [[#References|[a3]]]).
+
Let $  \Omega $
 +
be a compact Hausdorff space, $  C( \Omega ) $
 +
the space of continuous functions on $  \Omega $
 +
with the sup-norm, and $  T : C( \Omega ) \rightarrow X $
 +
a bounded linear operator ( $  X $
 +
is any Banach space). Then $  T $
 +
can be represented by a weak- $  * $
 +
countably-additive vector measure $  F $
 +
defined on the $  \sigma $-
 +
field of Borel sets in $  \Omega $
 +
and taking its values in $  X  ^ {**} $,  
 +
the bidual of $  X $(
 +
cf. [[Adjoint space|Adjoint space]]). This representation is particularly satisfactory when $  T $
 +
is weakly compact, for then $  F $
 +
has its values in $  X $,  
 +
and is countably additive (either of these properties is in fact equivalent to $  T $
 +
being weakly compact). Then one has $  T f = \int _  \Omega  f  dF $(
 +
$  f \in C( \Omega ) $),  
 +
where the integral has its more or less obvious meaning. An immediate consequence of this representation formula is that $  T $
 +
maps weakly-compact sets into norm-compact sets ( $  C( \Omega ) $
 +
has the Dunford–Pettis property). Other classes of operators $  T : C( \Omega ) \rightarrow X $
 +
such as the compact, the nuclear and the absolutely summing ones admit equally nice characterizations in terms of their representing measures (see [[#References|[a3]]]).
  
Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649084.png" /> be a bounded linear operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649085.png" /> into a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649086.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649087.png" /> a finite [[Measure space|measure space]]). There is an obvious vector measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649088.png" /> associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649089.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649090.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649091.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649092.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649093.png" />-continuous and of bounded variation. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649094.png" /> has a Radon–Nikodým derivative, i.e. if there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649095.png" />-valued Bochner-integrable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649096.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649097.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649098.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v09649099.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490100.png" /> can be represented as a [[Bochner integral|Bochner integral]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490101.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490102.png" />). It is known, however, that in general such a derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490103.png" /> does not exist. If, for a particular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490104.png" /> and for any measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490105.png" />, every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490106.png" />-continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490107.png" />-valued measure of bounded variation has a Radon–Nikodým derivative, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490108.png" /> is said to have the Radon–Nikodým property (RNP). Examples of spaces with the RNP: separable dual spaces (the Dunford–Pettis theorem) and reflexive spaces, so in particular Hilbert spaces. The spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490109.png" /> (i.e. the space of null sequences) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490110.png" /> fail the RNP. The RNP for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490111.png" /> has been shown to be equivalent to various convergence properties for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490113.png" />-valued martingales. In turn, this martingale approach has led to various purely geometrical characterizations of spaces with the RNP (see [[#References|[a1]]] for details). An example is as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490114.png" /> has the RNP if and only if for every closed bounded convex subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490115.png" /> and every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490116.png" /> there is a closed hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490117.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490118.png" /> so that both half-spaces determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490119.png" /> intersect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490120.png" />, and one of these intersections has diameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490121.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490122.png" /> is dentable). The Krein–Milman property states that every closed bounded convex set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490123.png" /> is the norm-closed hull of its extreme points. If a Banach space possesses the RNP, then it has the Krein–Milman property (J. Lindenstrauss). For dual spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490124.png" /> these two properties are equivalent.
+
Now, let $  T $
 +
be a bounded linear operator from $  L _ {1} ( \Omega , \Sigma , \mu ) $
 +
into a Banach space $  X $(
 +
( \Omega , \Sigma , \mu ) $
 +
a finite [[Measure space|measure space]]). There is an obvious vector measure $  F $
 +
associated to $  T $:  
 +
$  F( E) = T ( \chi _ {E} ) $,  
 +
$  E \in \Sigma $.  
 +
Moreover, $  F $
 +
is $  \mu $-
 +
continuous and of bounded variation. If $  F $
 +
has a Radon–Nikodým derivative, i.e. if there exists an $  X $-
 +
valued Bochner-integrable function $  f $
 +
on $  \Omega $
 +
such that $  F( E) = \int _ {E} f  d \mu $(
 +
$  E \in \Sigma $),  
 +
then $  T $
 +
can be represented as a [[Bochner integral|Bochner integral]]: $  Tg = \int _  \Omega  g f  d \mu $(
 +
$  g \in L _ {1} ( \mu ) $).  
 +
It is known, however, that in general such a derivative $  f $
 +
does not exist. If, for a particular $  X $
 +
and for any measure space $  ( \Omega , \Sigma , \mu ) $,  
 +
every $  \mu $-
 +
continuous $  X $-
 +
valued measure of bounded variation has a Radon–Nikodým derivative, then $  X $
 +
is said to have the Radon–Nikodým property (RNP). Examples of spaces with the RNP: separable dual spaces (the Dunford–Pettis theorem) and reflexive spaces, so in particular Hilbert spaces. The spaces $  c _ {0} $(
 +
i.e. the space of null sequences) and $  L _ {1} [ 0, 1] $
 +
fail the RNP. The RNP for $  X $
 +
has been shown to be equivalent to various convergence properties for $  X $-
 +
valued martingales. In turn, this martingale approach has led to various purely geometrical characterizations of spaces with the RNP (see [[#References|[a1]]] for details). An example is as follows: $  X $
 +
has the RNP if and only if for every closed bounded convex subset $  B \subset  X $
 +
and every $  \epsilon > 0 $
 +
there is a closed hyperplane $  H $
 +
in $  X $
 +
so that both half-spaces determined by $  H $
 +
intersect $  B $,  
 +
and one of these intersections has diameter < \epsilon $(
 +
$  X $
 +
is dentable). The Krein–Milman property states that every closed bounded convex set of $  X $
 +
is the norm-closed hull of its extreme points. If a Banach space possesses the RNP, then it has the Krein–Milman property (J. Lindenstrauss). For dual spaces $  X  ^ {*} $
 +
these two properties are equivalent.
  
The question can also be asked which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490125.png" />-continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096490/v096490126.png" />-valued measures are Pettis integrals (rather than Bochner integrals, cf. [[Pettis integral|Pettis integral]]). This leads to the so-called weak Radon–Nikodým property (WRNP) (see [[#References|[a6]]]).
+
The question can also be asked which $  \mu $-
 +
continuous $  X $-
 +
valued measures are Pettis integrals (rather than Bochner integrals, cf. [[Pettis integral|Pettis integral]]). This leads to the so-called weak Radon–Nikodým property (WRNP) (see [[#References|[a6]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.D. Bourgin,  "Geometric aspects of convex sets with the Radon–Nikodým property" , ''Lect. notes in math.'' , '''993''' , Springer  (1983)  {{MR|704815}} {{ZBL|0512.46017}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Dinculeanu,  "Vector measures" , Pergamon  (1967)  {{MR|0214722}} {{MR|0206190}} {{ZBL|0992.28006}} {{ZBL|0691.60030}} {{ZBL|0647.60062}} {{ZBL|0283.60051}} {{ZBL|0195.34002}} {{ZBL|0142.10502}} {{ZBL|0178.17302}} {{ZBL|0171.01701}} {{ZBL|0117.33702}} {{ZBL|0271.28006}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Diestel,  J.J. Uhl jr.,  "Vector measures" , ''Math. Surveys'' , '''15''' , Amer. Math. Soc.  (1977)  {{MR|0453964}} {{ZBL|0369.46039}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)  {{MR|0117523}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  I. Kluvanek,  G. Knowles,  "Vector measures and control systems" , North-Holland  (1975)  {{MR|0499068}} {{ZBL|0316.46043}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M. Talagrand,  "Pettis integral and measure theory"  ''Mem. Amer. Math. Soc.'' , '''307'''  (1984)  {{MR|0756174}} {{ZBL|0582.46049}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  G.E.F. Thomas,  "The Lebesgue–Nikodým theorem for vector valued Radon measures"  ''Mem. Amer. Math. Soc.'' , '''139'''  (1974)  {{MR|}} {{ZBL|0282.28004}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.D. Bourgin,  "Geometric aspects of convex sets with the Radon–Nikodým property" , ''Lect. notes in math.'' , '''993''' , Springer  (1983)  {{MR|704815}} {{ZBL|0512.46017}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Dinculeanu,  "Vector measures" , Pergamon  (1967)  {{MR|0214722}} {{MR|0206190}} {{ZBL|0992.28006}} {{ZBL|0691.60030}} {{ZBL|0647.60062}} {{ZBL|0283.60051}} {{ZBL|0195.34002}} {{ZBL|0142.10502}} {{ZBL|0178.17302}} {{ZBL|0171.01701}} {{ZBL|0117.33702}} {{ZBL|0271.28006}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Diestel,  J.J. Uhl jr.,  "Vector measures" , ''Math. Surveys'' , '''15''' , Amer. Math. Soc.  (1977)  {{MR|0453964}} {{ZBL|0369.46039}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)  {{MR|0117523}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  I. Kluvanek,  G. Knowles,  "Vector measures and control systems" , North-Holland  (1975)  {{MR|0499068}} {{ZBL|0316.46043}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M. Talagrand,  "Pettis integral and measure theory"  ''Mem. Amer. Math. Soc.'' , '''307'''  (1984)  {{MR|0756174}} {{ZBL|0582.46049}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  G.E.F. Thomas,  "The Lebesgue–Nikodým theorem for vector valued Radon measures"  ''Mem. Amer. Math. Soc.'' , '''139'''  (1974)  {{MR|}} {{ZBL|0282.28004}} </TD></TR></table>

Latest revision as of 09:13, 6 January 2024


A finitely-additive set function $ F $ defined on a field of subsets $ {\mathcal F} $ of a set $ \Omega $, with values in a Banach space $ X $( or, more generally, a topological vector space). A vector measure $ F $ is called strongly additive if $ \sum _ {n=1} ^ \infty F( E _ {n} ) $ converges in $ X $ for every sequence of pairwise disjoint sets $ E _ {n} \in {\mathcal F} $, and countably additive if, in addition, $ \sum _ {n=1} ^ \infty F( E _ {n} ) = F ( \cup _ {n=1} ^ \infty E _ {n} ) $ whenever $ \cup _ {n=1} ^ \infty E _ {n} $ belongs to $ {\mathcal F} $. If $ x ^ {*} F $ is countably additive for every $ x ^ {*} \in X ^ {*} $, then $ F $ is said to be weakly countably additive. A weakly countably-additive vector measure defined on a $ \sigma $- field is countably additive (the Orlicz–Pettis theorem). The variation $ | F | $ of $ F $ is the extended real-valued non-negative finitely-additive set function defined by

$$ | F | ( E) = \sup _ \pi \ \sum _ {A \in \pi } \| F( A) \| ,\ \ E \in {\mathcal F} , $$

where the supremum is over all finite partitions $ \pi $ of $ E $ into disjoint members of $ {\mathcal F} $. $ F $ is said to have bounded variation if $ | F | ( \Omega ) < \infty $. $ | F | $ is countably additive if and only if $ F $ is. The semi-variation $ \| F \| $ of $ F $ is defined by

$$ \| F \| ( E) = \sup \{ {| x ^ {*} F | ( E) } : { \| x ^ {*} \| \leq 1 } \} ,\ \ E \in {\mathcal F} . $$

$ \| F \| $ is a monotone finitely-subadditive set function, and if $ \| F \| ( \Omega ) < \infty $, then $ F $ is said to have bounded semi-variation. Since, equivalently, this means that the range of $ F $ is norm bounded, such measures are also called bounded. Vector measures of bounded variation are strongly additive, and strongly-additive vector measures are bounded. A bounded vector measure is strongly additive if and only if its range is relatively weakly compact. In particular, a countably-additive vector measure has relatively weakly-compact range.

Let $ ( F _ {n} ) $ be a sequence of $ X $- valued countably-additive vector measures defined on a $ \sigma $- field $ \Sigma $, and let each $ F _ {n} $ be $ \mu $- continuous, i.e. $ \lim\limits _ {\mu ( E) \rightarrow 0 } F _ {n} ( E) = 0 $, where $ \mu $ is a non-negative extended real-valued measure. Now, if $ \lim\limits _ {n \rightarrow \infty } F _ {n} ( E) = F( E) $ exists for every $ E \in \Sigma $, then the $ \mu $- continuity is uniform for $ n \in \mathbf N $, i.e. $ \lim\limits _ {\mu ( E) \rightarrow 0 } F _ {n} ( E) = 0 $, uniformly in $ n $. Hence $ F $ is $ \mu $- continuous. In particular, if $ \mu $ is finite it follows that $ F $ is countably additive. This is the Vitali–Hahn–Saks theorem. Another striking result from the theory of vector measures is the so-called Nikodým boundedness theorem: For a collection $ M $ of bounded vector measures $ F $ on a $ \sigma $- field $ \Sigma $, if $ \sup _ {F \in M } \| F( E) \| < \infty $ for each $ E \in \Sigma $, then $ M $ is uniformly bounded, i.e. $ \sup _ {F \in M , E \in \Sigma } \| F( E) \| < \infty $. There are also versions for strongly-additive vector measures of the well-known decomposition theorems of Yosida–Hewitt and of Lebesgue (see [a3]). Finally, a non-atomic $ X $- valued measure on a $ \sigma $- field has compact and convex range if $ \mathop{\rm dim} X < \infty $. This is Lyapunov's theorem. It fails for infinite-dimensional $ X $.

Vector measure theory has important applications to other areas of functional analysis. First of all to operator theory, where problems of representing operators on certain function spaces may well have been the original motive for studying vector measures. Much later, in the 1970s, the problem of differentiating vector measures led to a body of results in the geometry of Banach spaces, centering around the so-called Radon–Nikodým property. Below these developments are given briefly (see also [a1] and [a4]); see [a5] for the role of vector measures in control theory.

Let $ \Omega $ be a compact Hausdorff space, $ C( \Omega ) $ the space of continuous functions on $ \Omega $ with the sup-norm, and $ T : C( \Omega ) \rightarrow X $ a bounded linear operator ( $ X $ is any Banach space). Then $ T $ can be represented by a weak- $ * $ countably-additive vector measure $ F $ defined on the $ \sigma $- field of Borel sets in $ \Omega $ and taking its values in $ X ^ {**} $, the bidual of $ X $( cf. Adjoint space). This representation is particularly satisfactory when $ T $ is weakly compact, for then $ F $ has its values in $ X $, and is countably additive (either of these properties is in fact equivalent to $ T $ being weakly compact). Then one has $ T f = \int _ \Omega f dF $( $ f \in C( \Omega ) $), where the integral has its more or less obvious meaning. An immediate consequence of this representation formula is that $ T $ maps weakly-compact sets into norm-compact sets ( $ C( \Omega ) $ has the Dunford–Pettis property). Other classes of operators $ T : C( \Omega ) \rightarrow X $ such as the compact, the nuclear and the absolutely summing ones admit equally nice characterizations in terms of their representing measures (see [a3]).

Now, let $ T $ be a bounded linear operator from $ L _ {1} ( \Omega , \Sigma , \mu ) $ into a Banach space $ X $( $ ( \Omega , \Sigma , \mu ) $ a finite measure space). There is an obvious vector measure $ F $ associated to $ T $: $ F( E) = T ( \chi _ {E} ) $, $ E \in \Sigma $. Moreover, $ F $ is $ \mu $- continuous and of bounded variation. If $ F $ has a Radon–Nikodým derivative, i.e. if there exists an $ X $- valued Bochner-integrable function $ f $ on $ \Omega $ such that $ F( E) = \int _ {E} f d \mu $( $ E \in \Sigma $), then $ T $ can be represented as a Bochner integral: $ Tg = \int _ \Omega g f d \mu $( $ g \in L _ {1} ( \mu ) $). It is known, however, that in general such a derivative $ f $ does not exist. If, for a particular $ X $ and for any measure space $ ( \Omega , \Sigma , \mu ) $, every $ \mu $- continuous $ X $- valued measure of bounded variation has a Radon–Nikodým derivative, then $ X $ is said to have the Radon–Nikodým property (RNP). Examples of spaces with the RNP: separable dual spaces (the Dunford–Pettis theorem) and reflexive spaces, so in particular Hilbert spaces. The spaces $ c _ {0} $( i.e. the space of null sequences) and $ L _ {1} [ 0, 1] $ fail the RNP. The RNP for $ X $ has been shown to be equivalent to various convergence properties for $ X $- valued martingales. In turn, this martingale approach has led to various purely geometrical characterizations of spaces with the RNP (see [a1] for details). An example is as follows: $ X $ has the RNP if and only if for every closed bounded convex subset $ B \subset X $ and every $ \epsilon > 0 $ there is a closed hyperplane $ H $ in $ X $ so that both half-spaces determined by $ H $ intersect $ B $, and one of these intersections has diameter $ < \epsilon $( $ X $ is dentable). The Krein–Milman property states that every closed bounded convex set of $ X $ is the norm-closed hull of its extreme points. If a Banach space possesses the RNP, then it has the Krein–Milman property (J. Lindenstrauss). For dual spaces $ X ^ {*} $ these two properties are equivalent.

The question can also be asked which $ \mu $- continuous $ X $- valued measures are Pettis integrals (rather than Bochner integrals, cf. Pettis integral). This leads to the so-called weak Radon–Nikodým property (WRNP) (see [a6]).

References

[a1] R.D. Bourgin, "Geometric aspects of convex sets with the Radon–Nikodým property" , Lect. notes in math. , 993 , Springer (1983) MR704815 Zbl 0512.46017
[a2] N. Dinculeanu, "Vector measures" , Pergamon (1967) MR0214722 MR0206190 Zbl 0992.28006 Zbl 0691.60030 Zbl 0647.60062 Zbl 0283.60051 Zbl 0195.34002 Zbl 0142.10502 Zbl 0178.17302 Zbl 0171.01701 Zbl 0117.33702 Zbl 0271.28006
[a3] J. Diestel, J.J. Uhl jr., "Vector measures" , Math. Surveys , 15 , Amer. Math. Soc. (1977) MR0453964 Zbl 0369.46039
[a4] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523
[a5] I. Kluvanek, G. Knowles, "Vector measures and control systems" , North-Holland (1975) MR0499068 Zbl 0316.46043
[a6] M. Talagrand, "Pettis integral and measure theory" Mem. Amer. Math. Soc. , 307 (1984) MR0756174 Zbl 0582.46049
[a7] G.E.F. Thomas, "The Lebesgue–Nikodým theorem for vector valued Radon measures" Mem. Amer. Math. Soc. , 139 (1974) Zbl 0282.28004
How to Cite This Entry:
Vector measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_measure&oldid=28279
This article was adapted from an original article by D. van Dulst (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article