Perfect measure
A concept introduced by B.V. Gnedenko and A.N. Kolmogorov in [1] with the aim of "attaining a full harmony between abstract measure theory and measure theory in metric spaces" . The subsequent development of the theory has revealed other aspects of the value of this concept. On the one hand the class of perfect measures is very wide, and on the other, a number of unpleasant technical complications that occur in general measure theory do not arise if one restricts to perfect measures.
A finite measure on a
-algebra
of subsets of a set
is called perfect if for any real-valued measurable function
on
and any set
such that
,
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where is the class of open subsets of
. For
to be perfect, it is necessary that for any real-valued measurable function
on
there exists a Borel set
such that
, and sufficient that for any real-valued measurable function
on
and any set
for which
there exists a Borel set
such that
![]() |
Every discrete measure is perfect. A measure defined on a -algebra of subsets of a separable metric space that contains all open sets is perfect if and only if the measure of any measurable set is the least upper bound of the measures of its compact subsets. The restriction of a perfect measure
defined on
to any
-subalgebra of
is perfect. A measure induced by a perfect measure
on any subset
with
is perfect. The image of a perfect measure
under a measurable mapping of
into another measurable space is perfect. A measure is perfect if and only if its completion is perfect. For every measure on any
-subalgebra of a
-algebra
of subsets of a set
to be perfect it is necessary and sufficient that for any real-valued
-measurable function
the set
is universally measurable (that is, it belongs to the domain of definition of the completion of every Borel measure on
). If
and if
is the
-algebra of Borel subsets of
, then every measure on
is perfect if and only if
is universally measurable.
Every space with a perfect measure such that
has a countable numbers of generators
separating points of
(that is, for all
,
, there is an
:
,
or
,
) is almost isomorphic to some space
, consisting of the Lebesgue measure on a finite interval and of a countable sequence (possibly empty) of points of positive mass (i.e., there is an
with
and a one-to-one mapping
of
onto
such that
and
are measurable and
).
Let be any index set and let
be a given space with a perfect measure for each
. Put
and let
be the algebra generated by the class of sets of the form
. If
is a finitely-additive measure on
such that
for all
and
, then: 1)
is countably additive on
; and 2) the extension
of
to the
-algebra
generated by
is perfect.
Let be a space with a perfect probability measure and let
,
be two
-subalgebras of the
-algebra
, where
has a countable number of generators. Then there is a regular conditional probability on
given
, i.e. there is a function
on
such that: 1) for a fixed
,
is a probability measure on
; 2) for a fixed
,
is measurable with respect to
; and 3)
for all
and
. Moreover, the function
can be chosen in such a way that the measures
are perfect. Let
,
be two measurable spaces and let
be a transition probability on
, that is,
is measurable with respect to
and
is a probability measure on
for all
,
. If the
are discrete and
is a perfect probability measure on
, then the measure
is perfect.
Perfect measures are closely connected with compact measures. A class of subsets of
is called compact if
,
and
implies that
for some
. A finite measure
on
is called compact if there is a compact class
such that for all
and
one can choose a
and an
such that
and
. Every compact measure is perfect. For a measure to be perfect it is necessary and sufficient that its restriction to any
-subalgebra with a countable number of generators be compact.
References
[1] | B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian) |
[2] | E. Marczewski, "On compact measures" Fund. Math. , 40 (1953) pp. 113–124 |
[3] | C. Ryll-Nardzewski, "On quasi-compact measures" Fund. Math. , 40 (1953) pp. 125–130 |
[4] | V.V. Sazonov, "On perfect measures" Transl. Amer. Math. Soc. (2) , 48 (1965) pp. 229–254 Izv. Akad. Nauk SSSR Ser. Mat. , 26 (1962) pp. 391–414 |
[5] | D. Ramachandran, "Perfect measures" , 1–2 , Macmillan (1979) |
Comments
References
[a1] | Kia-An Yen, "Forme mesurable de la théorie des ensembles sousliniens, applications à la théorie de la mesure" Scientia Sinica , XVIII (1975) pp. 444–463 |
Perfect measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Perfect_measure&oldid=13542