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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=14783
This article was adapted from an original article by I.M. Bomze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article