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An order-complete [[Banach lattice|Banach lattice]] (cf. also [[Riesz space|Riesz space]]) of measures on a [[Measurable space|measurable space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l1100302.png" />, defined in the context of [[Statistical decision theory|statistical decision theory]] [[#References|[a2]]], [[#References|[a5]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a10]]]. Prime object of this theory is the statistical experiment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l1100303.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l1100304.png" /> is a set of probability measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l1100305.png" />. A statistical decision problem is to determine which of the distributions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l1100306.png" /> are most likely to generate the observations (or data) collected. While the [[Radon–Nikodým theorem|Radon–Nikodým theorem]] guarantees that one can operate with densities
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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/l/l110/l110030/l1100307.png" /></td> </tr></table>
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of distributions if all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l1100308.png" /> are dominated by a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l1100309.png" />-finite measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003010.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003011.png" />, there is no such possibility in the undominated case. Nevertheless, there is a substitute for the space generated by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003012.png" /> which respects both the linear and the order structure of measures: the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003013.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003014.png" /> of the experiment, introduced in [[#References|[a4]]]. This is a subspace of the Banach lattice of all signed measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003015.png" />, and can be defined in three different ways, as follows [[#References|[a1]]].
+
An order-complete [[Banach lattice|Banach lattice]] (cf. also [[Riesz space|Riesz space]]) of measures on a [[Measurable space|measurable space]]    ( \Omega, {\mathcal F} ) ,  
 +
defined in the context of [[Statistical decision theory|statistical decision theory]] [[#References|[a2]]], [[#References|[a5]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a10]]]. Prime object of this theory is the statistical experiment    {\mathcal E} = ( \Omega, {\mathcal F}, {\mathcal P} )
 +
where    {\mathcal P}
 +
is a set of probability measures on   ( \Omega, {\mathcal F} ) .  
 +
A statistical decision problem is to determine which of the distributions in    {\mathcal P}
 +
are most likely to generate the observations (or data) collected. While the [[Radon–Nikodým theorem|Radon–Nikodým theorem]] guarantees that one can operate with densities
  
Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003016.png" /> the vector lattice of all signed finite measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003017.png" />, put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003018.png" /> and use <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003019.png" /> as an abbreviation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003020.png" />. Equipped with the variational norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003022.png" /> is an order-complete Banach lattice. More precisely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003023.png" /> is an abstract <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003025.png" />-space, which means that the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003026.png" /> is additive on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003027.png" />. A solid linear subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003028.png" /> is called a band if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003029.png" /> whenever the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003030.png" /> satisfy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003031.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003032.png" />.
+
$$
 +
{
 +
\frac{dP }{d \mu }
 +
} \in L _ {1} ( \mu )
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003033.png" /> is a statistical experiment, then one defines
+
of distributions if all    P \in {\mathcal P}
 +
are dominated by a    \sigma -
 +
finite measure    \mu
 +
on    ( \Omega, {\mathcal F} ) ,
 +
there is no such possibility in the undominated case. Nevertheless, there is a substitute for the space generated by the  $  { {dP } / {d \mu } } $
 +
which respects both the linear and the order structure of measures: the    L -
 +
space    L ( {\mathcal E} )
 +
of the experiment, introduced in [[#References|[a4]]]. This is a subspace of the Banach lattice of all signed measures on    ( \Omega, {\mathcal F} ) ,
 +
and can be defined in three different ways, as follows [[#References|[a1]]].
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003034.png" /> to be the smallest band (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003035.png" />) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003036.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003037.png" />;
+
Denote by  $  { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) $
 +
the vector lattice of all signed finite measures on  $  ( \Omega, {\mathcal F} ) $,
 +
put  $  | \mu | = \sup  ( \mu, - \mu ) \in { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) $
 +
and use    \mu \perp  \nu
 +
as an abbreviation for  $  \inf  ( | \mu | , | \nu | ) = 0 $.  
 +
Equipped with the variational norm  $  \| \mu \| = | \mu | ( \Omega ) $,
 +
  { \mathop{\rm ca} } ( \Omega, {\mathcal F} )
 +
is an order-complete Banach lattice. More precisely,    { \mathop{\rm ca} } ( \Omega, {\mathcal F} )
 +
is an abstract    L -
 +
space, which means that the norm    \| \cdot \|
 +
is additive on    { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) _ {+} .  
 +
A solid linear subspace    {\mathcal D} \subseteq { \mathop{\rm ca} } ( \Omega, {\mathcal F} )
 +
is called a band if    \sup  _ {i \in I }  \mu _ {i} \in {\mathcal D}
 +
whenever the    \mu _ {i} \in {\mathcal D}
 +
satisfy    \mu _ {i} \leq  \mu \in { \mathop{\rm ca} } ( \Omega, {\mathcal F} )
 +
for all    i \in I .
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003038.png" /> to be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003039.png" />-closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003040.png" />, where
+
If  $  {\mathcal E} = ( \Omega, {\mathcal F}, {\mathcal P} ) $
 +
is a statistical experiment, then one defines
  
<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/l/l110/l110030/l11003041.png" /></td> </tr></table>
+
a)    L _ {1} ( {\mathcal E} )
 +
to be the smallest band (with respect to    \subseteq )
 +
in    { \mathop{\rm ca} } ( \Omega, {\mathcal F} )
 +
containing    {\mathcal P} ;
  
<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/l/l110/l110030/l11003042.png" /></td> </tr></table>
+
b)    L _ {2} ( {\mathcal E} )
 +
to be the    \| \cdot \| -
 +
closure of    L  ^  \prime  ( {\mathcal E} ) ,
 +
where
  
c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003043.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003044.png" />. This space is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003046.png" />-space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003047.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003048.png" />.
+
$$
 +
L  ^  \prime  ( {\mathcal E} ) = \left \{ {\mu \in { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) } : \left | \mu \right | \leq  \sum _ {i = 1 } ^ { n }  \alpha _ {i} P _ {i} \right .  
 +
$$
  
If there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003049.png" /> such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003050.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003051.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003052.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003053.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003054.png" /> is dominated (and vice versa). In this case, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003055.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003056.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003057.png" /> is, as a Banach lattice, isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003058.png" />. The situation for undominated experiments is different. As an abstract <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003059.png" />-space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003060.png" /> is always isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003061.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003062.png" /> a [[Radon measure|Radon measure]] on a locally compact topological space [[#References|[a3]]]. However, in general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003063.png" /> is not even semi-finite [[#References|[a6]]] (i.e., lacks the finite subset property [[#References|[a11]]]), and then there is no representation of the topological dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003064.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003065.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003066.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003068.png" />-space of the experiment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003069.png" /> and generalizes the space of equivalence classes of bounded random variables in the following sense. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003070.png" /> denote the set of all real-valued functions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003071.png" /> that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003072.png" />-measurable and bounded. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003073.png" />, denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003074.png" /> the mapping assigning <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003075.png" /> to every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003076.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003077.png" /> coincides with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003078.png" />-closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003079.png" /> [[#References|[a1]]], [[#References|[a4]]], [[#References|[a8]]]. For an alternative representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003080.png" />, see [[#References|[a9]]].
+
$$
 +
\left .
 +
{\textrm{ for some  }  P _ {i} \in {\mathcal P}, \alpha _ {i} \geq  0, \textrm{ all  }  i ; n \in \mathbf N } \right \} ;
 +
$$
  
An experiment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003081.png" /> is called coherent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003082.png" />. Every dominated experiment is also coherent, due to the familiar isomorphism between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003083.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003084.png" />, the reverse implication being false in general (for even larger classes of statistical experiments, see, e.g., [[#References|[a6]]]). However, every coherent experiment is weakly dominated (and vice versa) in the following sense [[#References|[a7]]]: there exists a semi-finite (not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003085.png" />-finite, in general) and localizable [[#References|[a11]]] measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003086.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003087.png" /> such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003088.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003089.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003090.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003091.png" />. This result is an alternative interpretation of the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003092.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003093.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003094.png" /> is semi-finite and localizable [[#References|[a11]]].
+
c) $  L _ {3} ( {\mathcal E} ) = \{ {\mu \in { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) } : {\mu \perp  \sigma  \textrm{ for all  }  \sigma \perp  {\mathcal P} } \} $.  
 +
Then  $  L _ {1} ( {\mathcal E} ) = L _ {2} ( {\mathcal E} ) = L _ {3} ( {\mathcal E} ) $.  
 +
This space is called the    L -
 +
space of   {\mathcal E}
 +
and is denoted by    L ( {\mathcal E} ) .
  
The experiment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003095.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003096.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003097.png" /> the Borel field, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003098.png" /> is not coherent, since the counting measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l11003099.png" /> is not localizable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l110030100.png" /> because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l110030101.png" /> is countably generated but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l110030102.png" /> is not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l110030103.png" />-finite [[#References|[a6]]] (this argument needs the assumption that each uncountable metric space contains a non-Borel set).
+
If there exists a    Q \in { \mathop{\rm ca} } ( \Omega, {\mathcal F} )
 +
such that for    A \in {\mathcal F}
 +
one has  $  Q ( A ) = 0 $
 +
if and only if  $  P ( A ) = 0 $
 +
for all    P \in {\mathcal P} ,
 +
then    {\mathcal E}
 +
is dominated (and vice versa). In this case, the    L -
 +
space    L ( {\mathcal E} )
 +
of    {\mathcal E}
 +
is, as a Banach lattice, isomorphic to    L  ^ {1} ( Q ) .  
 +
The situation for undominated experiments is different. As an abstract    L -
 +
space,    L ( {\mathcal E} )
 +
is always isomorphic to    L  ^ {1} ( m ) ,
 +
with   m
 +
a [[Radon measure|Radon measure]] on a locally compact topological space [[#References|[a3]]]. However, in general    m
 +
is not even semi-finite [[#References|[a6]]] (i.e., lacks the finite subset property [[#References|[a11]]]), and then there is no representation of the topological dual  $  M ( {\mathcal E} ) = L ( {\mathcal E} )  ^ {*} $
 +
as    L  ^  \infty  ( m ) .
 +
  M ( {\mathcal E} )
 +
is called the    M -
 +
space of the experiment    {\mathcal E}
 +
and generalizes the space of equivalence classes of bounded random variables in the following sense. Let    X
 +
denote the set of all real-valued functions defined on    \Omega
 +
that are    {\mathcal F} -
 +
measurable and bounded. For any    \varphi \in X ,
 +
denote by    {\dot \varphi  }
 +
the mapping assigning    \int _  \Omega  \varphi  {d \mu }
 +
to every    \mu \in L ( {\mathcal E} ) .  
 +
Then    M ( {\mathcal E} )
 +
coincides with the   \sigma ( M ( {\mathcal E} ) ,L ( {\mathcal E} ) ) -
 +
closure of    {\dot{X} } [[#References|[a1]]], [[#References|[a4]]], [[#References|[a8]]]. For an alternative representation of    M ( {\mathcal E} ) ,
 +
see [[#References|[a9]]].
 +
 
 +
An experiment    {\mathcal E}
 +
is called coherent if    M ( {\mathcal E} ) = {\dot{X} } .  
 +
Every dominated experiment is also coherent, due to the familiar isomorphism between    [ L  ^ {1} ( Q ) ]  ^ {*}
 +
and    L  ^  \infty  ( Q ) ,
 +
the reverse implication being false in general (for even larger classes of statistical experiments, see, e.g., [[#References|[a6]]]). However, every coherent experiment is weakly dominated (and vice versa) in the following sense [[#References|[a7]]]: there exists a semi-finite (not   \sigma -
 +
finite, in general) and localizable [[#References|[a11]]] measure    \mu
 +
on   ( \Omega, {\mathcal F} )
 +
such that for    A \in {\mathcal F}
 +
one has  $  \mu ( A ) = 0 $
 +
if and only if  $  P ( A ) = 0 $
 +
for all    P \in {\mathcal P} .  
 +
This result is an alternative interpretation of the fact that    L  ^  \infty  ( \mu )
 +
is isomorphic to    [ L  ^ {1} ( \mu ) ]  ^ {*}
 +
if and only if    \mu
 +
is semi-finite and localizable [[#References|[a11]]].
 +
 
 +
The experiment  $  {\mathcal E} = ( \Omega, {\mathcal F}, {\mathcal P} ) $
 +
with  $  \Omega = [ 0,1 ] $,
 +
  {\mathcal F}
 +
the Borel field, and  $  {\mathcal P} = \{ {\delta _ {x} } : {x \in [ 0,1 ] } \} $
 +
is not coherent, since the counting measure    \mu
 +
is not localizable on    {\mathcal F}
 +
because    {\mathcal F}
 +
is countably generated but   \mu
 +
is not   \sigma -
 +
finite [[#References|[a6]]] (this argument needs the assumption that each uncountable metric space contains a non-Borel set).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.M. Bomze,  "A functional analytic approach to statistical experiments" , Longman  (1990)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Heyer,  "Theory of statistical experiments" , Springer  (1982)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Kakutani,  "Concrete representation of abstract <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l110030104.png" />-spaces and the mean ergodic theorem"  ''Ann. of Math.'' , '''42'''  (1941)  pp. 523–537</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Le Cam,  "Sufficiency and approximate sufficiency"  ''Ann. Math. Stat.'' , '''35'''  (1964)  pp. 1419–1455</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L. Le Cam,  "Asymptotic methods in statistical decision theory" , Springer  (1986)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  H. Luschgy,  D. Mussmann,  "Products of majorized experiments"  ''Statistics and Decision'' , '''4'''  (1986)  pp. 321–335</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  E. Siebert,  "Pairwise sufficiency"  ''Z. Wahrscheinlichkeitsth. verw. Gebiete'' , '''46'''  (1979)  pp. 237–246</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  H. Strasser,  "Mathematical theory of statistics" , de Gruyter  (1985)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  E.N. Torgersen,  "On complete sufficient statistics and uniformly minimum variance unbiased estimators"  ''Teoria statistica delle decisioni. Symp. Math.'' , '''25'''  (1980)  pp. 137–153</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  E.N. Torgersen,  "Comparison of statistical experiments" , Cambridge Univ. Press  (1991)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  A.C. Zaanen,  "Integration" , North-Holland  (1967)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.M. Bomze,  "A functional analytic approach to statistical experiments" , Longman  (1990)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Heyer,  "Theory of statistical experiments" , Springer  (1982)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Kakutani,  "Concrete representation of abstract <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110030/l110030104.png" />-spaces and the mean ergodic theorem"  ''Ann. of Math.'' , '''42'''  (1941)  pp. 523–537</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Le Cam,  "Sufficiency and approximate sufficiency"  ''Ann. Math. Stat.'' , '''35'''  (1964)  pp. 1419–1455</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L. Le Cam,  "Asymptotic methods in statistical decision theory" , Springer  (1986)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  H. Luschgy,  D. Mussmann,  "Products of majorized experiments"  ''Statistics and Decision'' , '''4'''  (1986)  pp. 321–335</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  E. Siebert,  "Pairwise sufficiency"  ''Z. Wahrscheinlichkeitsth. verw. Gebiete'' , '''46'''  (1979)  pp. 237–246</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  H. Strasser,  "Mathematical theory of statistics" , de Gruyter  (1985)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  E.N. Torgersen,  "On complete sufficient statistics and uniformly minimum variance unbiased estimators"  ''Teoria statistica delle decisioni. Symp. Math.'' , '''25'''  (1980)  pp. 137–153</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  E.N. Torgersen,  "Comparison of statistical experiments" , Cambridge Univ. Press  (1991)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  A.C. Zaanen,  "Integration" , North-Holland  (1967)</TD></TR></table>

Revision as of 22:15, 5 June 2020


An order-complete Banach lattice (cf. also Riesz space) of measures on a measurable space ( \Omega, {\mathcal F} ) , defined in the context of statistical decision theory [a2], [a5], [a7], [a8], [a10]. Prime object of this theory is the statistical experiment {\mathcal E} = ( \Omega, {\mathcal F}, {\mathcal P} ) where {\mathcal P} is a set of probability measures on ( \Omega, {\mathcal F} ) . A statistical decision problem is to determine which of the distributions in {\mathcal P} are most likely to generate the observations (or data) collected. While the Radon–Nikodým theorem guarantees that one can operate with densities

{ \frac{dP }{d \mu } } \in L _ {1} ( \mu )

of distributions if all P \in {\mathcal P} are dominated by a \sigma - finite measure \mu on ( \Omega, {\mathcal F} ) , there is no such possibility in the undominated case. Nevertheless, there is a substitute for the space generated by the { {dP } / {d \mu } } which respects both the linear and the order structure of measures: the L - space L ( {\mathcal E} ) of the experiment, introduced in [a4]. This is a subspace of the Banach lattice of all signed measures on ( \Omega, {\mathcal F} ) , and can be defined in three different ways, as follows [a1].

Denote by { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) the vector lattice of all signed finite measures on ( \Omega, {\mathcal F} ) , put | \mu | = \sup ( \mu, - \mu ) \in { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) and use \mu \perp \nu as an abbreviation for \inf ( | \mu | , | \nu | ) = 0 . Equipped with the variational norm \| \mu \| = | \mu | ( \Omega ) , { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) is an order-complete Banach lattice. More precisely, { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) is an abstract L - space, which means that the norm \| \cdot \| is additive on { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) _ {+} . A solid linear subspace {\mathcal D} \subseteq { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) is called a band if \sup _ {i \in I } \mu _ {i} \in {\mathcal D} whenever the \mu _ {i} \in {\mathcal D} satisfy \mu _ {i} \leq \mu \in { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) for all i \in I .

If {\mathcal E} = ( \Omega, {\mathcal F}, {\mathcal P} ) is a statistical experiment, then one defines

a) L _ {1} ( {\mathcal E} ) to be the smallest band (with respect to \subseteq ) in { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) containing {\mathcal P} ;

b) L _ {2} ( {\mathcal E} ) to be the \| \cdot \| - closure of L ^ \prime ( {\mathcal E} ) , where

L ^ \prime ( {\mathcal E} ) = \left \{ {\mu \in { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) } : \left | \mu \right | \leq \sum _ {i = 1 } ^ { n } \alpha _ {i} P _ {i} \right .

\left . {\textrm{ for some } P _ {i} \in {\mathcal P}, \alpha _ {i} \geq 0, \textrm{ all } i ; n \in \mathbf N } \right \} ;

c) L _ {3} ( {\mathcal E} ) = \{ {\mu \in { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) } : {\mu \perp \sigma \textrm{ for all } \sigma \perp {\mathcal P} } \} . Then L _ {1} ( {\mathcal E} ) = L _ {2} ( {\mathcal E} ) = L _ {3} ( {\mathcal E} ) . This space is called the L - space of {\mathcal E} and is denoted by L ( {\mathcal E} ) .

If there exists a Q \in { \mathop{\rm ca} } ( \Omega, {\mathcal F} ) such that for A \in {\mathcal F} one has Q ( A ) = 0 if and only if P ( A ) = 0 for all P \in {\mathcal P} , then {\mathcal E} is dominated (and vice versa). In this case, the L - space L ( {\mathcal E} ) of {\mathcal E} is, as a Banach lattice, isomorphic to L ^ {1} ( Q ) . The situation for undominated experiments is different. As an abstract L - space, L ( {\mathcal E} ) is always isomorphic to L ^ {1} ( m ) , with m a Radon measure on a locally compact topological space [a3]. However, in general m is not even semi-finite [a6] (i.e., lacks the finite subset property [a11]), and then there is no representation of the topological dual M ( {\mathcal E} ) = L ( {\mathcal E} ) ^ {*} as L ^ \infty ( m ) . M ( {\mathcal E} ) is called the M - space of the experiment {\mathcal E} and generalizes the space of equivalence classes of bounded random variables in the following sense. Let X denote the set of all real-valued functions defined on \Omega that are {\mathcal F} - measurable and bounded. For any \varphi \in X , denote by {\dot \varphi } the mapping assigning \int _ \Omega \varphi {d \mu } to every \mu \in L ( {\mathcal E} ) . Then M ( {\mathcal E} ) coincides with the \sigma ( M ( {\mathcal E} ) ,L ( {\mathcal E} ) ) - closure of {\dot{X} } [a1], [a4], [a8]. For an alternative representation of M ( {\mathcal E} ) , see [a9].

An experiment {\mathcal E} is called coherent if M ( {\mathcal E} ) = {\dot{X} } . Every dominated experiment is also coherent, due to the familiar isomorphism between [ L ^ {1} ( Q ) ] ^ {*} and L ^ \infty ( Q ) , the reverse implication being false in general (for even larger classes of statistical experiments, see, e.g., [a6]). However, every coherent experiment is weakly dominated (and vice versa) in the following sense [a7]: there exists a semi-finite (not \sigma - finite, in general) and localizable [a11] measure \mu on ( \Omega, {\mathcal F} ) such that for A \in {\mathcal F} one has \mu ( A ) = 0 if and only if P ( A ) = 0 for all P \in {\mathcal P} . This result is an alternative interpretation of the fact that L ^ \infty ( \mu ) is isomorphic to [ L ^ {1} ( \mu ) ] ^ {*} if and only if \mu is semi-finite and localizable [a11].

The experiment {\mathcal E} = ( \Omega, {\mathcal F}, {\mathcal P} ) with \Omega = [ 0,1 ] , {\mathcal F} the Borel field, and {\mathcal P} = \{ {\delta _ {x} } : {x \in [ 0,1 ] } \} is not coherent, since the counting measure \mu is not localizable on {\mathcal F} because {\mathcal F} is countably generated but \mu is not \sigma - finite [a6] (this argument needs the assumption that each uncountable metric space contains a non-Borel set).

References

[a1] I.M. Bomze, "A functional analytic approach to statistical experiments" , Longman (1990)
[a2] H. Heyer, "Theory of statistical experiments" , Springer (1982)
[a3] S. Kakutani, "Concrete representation of abstract -spaces and the mean ergodic theorem" Ann. of Math. , 42 (1941) pp. 523–537
[a4] L. Le Cam, "Sufficiency and approximate sufficiency" Ann. Math. Stat. , 35 (1964) pp. 1419–1455
[a5] L. Le Cam, "Asymptotic methods in statistical decision theory" , Springer (1986)
[a6] H. Luschgy, D. Mussmann, "Products of majorized experiments" Statistics and Decision , 4 (1986) pp. 321–335
[a7] E. Siebert, "Pairwise sufficiency" Z. Wahrscheinlichkeitsth. verw. Gebiete , 46 (1979) pp. 237–246
[a8] H. Strasser, "Mathematical theory of statistics" , de Gruyter (1985)
[a9] E.N. Torgersen, "On complete sufficient statistics and uniformly minimum variance unbiased estimators" Teoria statistica delle decisioni. Symp. Math. , 25 (1980) pp. 137–153
[a10] E.N. Torgersen, "Comparison of statistical experiments" , Cambridge Univ. Press (1991)
[a11] A.C. Zaanen, "Integration" , North-Holland (1967)
How to Cite This Entry:
L-space-of-a-statistical-experiment. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L-space-of-a-statistical-experiment&oldid=47546
This article was adapted from an original article by I.M. Bomze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article