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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]]  $  ( \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

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

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 [[#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]]].

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|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====
<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>