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An analogue of the concept of a [[Characteristic function|characteristic function]]; it is used in the infinite-dimensional case. Let  $  \mathfrak X $
be a non-empty set, let  $  \Gamma $
be a vector space of real-valued functions defined on  $  \mathfrak X $
and let  $  \widehat{C}  ( \mathfrak X , \Gamma ) $
be the smallest  $  \sigma $-
algebra of subsets of  $  \mathfrak X $
relative to which all functions in  $  \Gamma $
are measurable. The characteristic functional of a probability measure  $  \mu $
given on  $  \widehat{C}  ( \mathfrak X , \Gamma ) $
is defined as the complex-valued functional  $  \widehat \mu   $
on  $  \Gamma $
given by the equation

$$ 
\widehat \mu   ( g)  = \ 
\int\limits _ {\mathfrak X }
 \mathop{\rm exp} [ ig ( x)]  d \mu ( x),\ \ 
g \in \Gamma .
$$

From now on the most important and simplest case when  $  \mathfrak X $
is a separable real Banach space and  $  \Gamma $
is its topological dual  $  \mathfrak X  ^ {*} $
is studied. In this case  $  \widehat{C}  ( \mathfrak X , \mathfrak X  ^ {*} ) $
coincides with the  $  \sigma $-
algebra of Borel sets of  $  \mathfrak X $. 
The concept of a characteristic functional for infinite-dimensional Banach spaces was introduced by A.N. Kolmogorov in [[#References|[1]]].

The characteristic functional of a random variable  $  X $
with values in  $  \mathfrak X $
is, by definition, that of its probability distribution  $  \mu _ {X} ( B) = {\mathsf P} \{ X \in B \} $, 
$  B \subset  \mathfrak X $.

Main properties of the characteristic functional:

1)  $  \widehat \mu   ( 0) = 1 $
and  $  \widehat \mu   $
is positive definite, i.e.  $  \sum _ {k,l} \alpha _ {k} \overline \alpha \; _ {l} \widehat \mu   ( x _ {k}  ^ {*} - x _ {l}  ^ {*} ) \geq  0 $
for any finite set of complex numbers  $  \alpha _ {i} $
and elements  $  x _ {i}  ^ {*} \in \mathfrak X  ^ {*} $;

2)  $  \widehat \mu   $
is continuous in the strong topology and sequentially continuous in the weak  $  * $
topology of  $  \mathfrak X  ^ {*} $;

3)  $  | \widehat \mu   ( x  ^ {*} ) | \leq  1 $,

$$ 
| \widehat \mu   ( x _ {1}  ^ {*} ) -
\widehat \mu   ( x _ {2}  ^ {*} ) |  ^ {2}  \leq  \ 
2 [ 1 -  \mathop{\rm Re}  \widehat \mu   ( x _ {1}  ^ {*} - x _ {2}  ^ {*} )],
$$

where  $  x  ^ {*} , x _ {1}  ^ {*} , x _ {2}  ^ {*} \in \mathfrak X  ^ {*} $;

4)  $  \overline{ {\widehat \mu   ( x  ^ {*} ) }}\; = \widehat \mu   (- x  ^ {*} ) $; 
in particular,  $  \widehat \mu   $
takes only real values (and is an even functional) if and only if the measure  $  \mu $
is symmetric, that is,  $  \mu ( B) = \mu (- B) $, 
where  $  - B = \{ {x } : {- x \in B } \} $;

5) the characteristic functional determines the measure uniquely;

6) the characteristic functional of the convolution of two probability measures (of the sum of two independent random variables) is the product of their characteristic functionals.

In the finite-dimensional case the method of characteristic functionals is based on the theorem about the continuity of the correspondence between measures and their characteristic functionals, and on a theorem concerning the description of the class of characteristic functionals. In the infinite-dimensional case the direct analogues of these theorems do not hold. If a sequence of probability measures  $  ( \mu _ {n} ) $
converges weakly to  $  \mu $, 
then  $  ( \widehat \mu   _ {n} ) $
converges pointwise to  $  \widehat \mu   $, 
and this convergence is uniform on bounded subsets of  $  \mathfrak X  ^ {*} $; 
if  $  K $
is a weakly relatively-compact family of probability measures on  $  \mathfrak X $, 
then the family  $  \{ {\widehat \mu   } : {\mu \in K } \} $
is equicontinuous in the strong topology of  $  \mathfrak X  ^ {*} $. 
The converse assertions only hold in the finite-dimensional case. However, the conditions of convergence and of weak relative compactness of families of probability measures can be expressed in terms of characteristic functionals (see [[#References|[2]]]). Furthermore, in contrast to the finite-dimensional case, not every positive-definite normalized (equal to 1 at the origin) continuous functional is a characteristic functional: continuity in the metric topology is not sufficient. A topology in  $  \mathfrak X  ^ {*} $
is called sufficient, or necessary, if in this topology the continuity of a positive-definite normalized functional is sufficient, or necessary, for it to be the characteristic functional of some probability measure on  $  \mathfrak X $. 
A necessary and sufficient topology is said to be an  $  S $-
topology. A space  $  \mathfrak X $
is called an  $  S $-
space if there is an  $  S $-
topology on  $  \mathfrak X  ^ {*} $. 
A Hilbert space is an  $  S $-
space (see [[#References|[3]]]).

The most important characteristic functionals are those of Gaussian measures. A measure  $  \mu $
in  $  \mathfrak X $
is called a centred Gaussian measure if for all  $  x  ^ {*} \in \mathfrak X  ^ {*} $,

$$ \tag{* }
\widehat \mu   ( x  ^ {*} )  = \ 
 \mathop{\rm exp} \left [ - {
\frac{1}{2}
 }
x  ^ {*} ( Rx  ^ {*} )
\right ] ,
$$

where  $  R $, 
a bounded linear positive operator from  $  \mathfrak X  ^ {*} $
into  $  \mathfrak X $, 
is the covariance operator of the measure  $  \mu $, 
defined by the relation

$$ 
x  ^ {*} ( Rx  ^ {*} )  = \ 
\int\limits x  ^ {*} 2 ( x)  d \mu ( x)
$$

(see [[#References|[4]]]). In contrast to the finite-dimensional case, not every functional of the form (*) is a characteristic functional: additional restrictions on  $  R $
are needed, depending on the space  $  \mathfrak X $. 
For example, if  $  \mathfrak X = l _ {p} $, 
$  1 \leq  p < \infty $, 
then an additional (necessary and sufficient) condition is  $  \sum r _ {kk}  ^ {p/2} < + \infty $, 
where  $  \| r _ {ij} \| $
is the matrix of the operator  $  R $
in the natural basis (see [[#References|[5]]]). In particular, in a Hilbert space the additional condition is that the operator  $  R $
be nuclear.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,   ''C.R. Acad. Sci. Paris'' , '''200'''  (1935)  pp. 1717–1718</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Yu.V. Prokhorov,   "Convergence of random processes and limit theorems in probability theory"  ''Theory Probab. Appl.'' , '''1'''  (1956)  pp. 157–214  ''Teor. Veroyatnost. i Primen.'' , '''1''' :  2  (1956)  pp. 177–238</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.V. Sazonov,   "A remark on characteristic functionals"  ''Theory Probab. Appl.'' , '''3'''  (1958)  pp. 188–192  ''Teor. Veroyatnost. i Primen.'' , '''3''' :  2  (1958)  pp. 201–205</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.N. Vakhania,   V.I. Tarieladze,   S.A. Chobanyan,   "Probability distributions on Banach spaces" , Reidel  (1987)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.N. Vakhania,   "Sur les répartitions de probabilités dans les espaces de suites numériques"  ''C.R. Acad. Sci. Paris'' , '''260'''  (1965)  pp. 1560–1562</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.N. Vakhania,   "Probability distributions on linear spaces" , North-Holland  (1981)  (Translated from Russian)</TD></TR></table>