Difference between revisions of "Vector measure"
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.D. Bourgin, "Geometric aspects of convex sets with the Radon–Nikodým property" , ''Lect. notes in math.'' , '''993''' , Springer (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Dinculeanu, "Vector measures" , Pergamon (1967)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Diestel, J.J. Uhl jr., "Vector measures" , ''Math. Surveys'' , '''15''' , Amer. Math. Soc. (1977)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> N. Dunford, J.T. Schwartz, "Linear operators. General theory" , '''1''' , Interscience (1958)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> I. Kluvanek, G. Knowles, "Vector measures and control systems" , North-Holland (1975)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> M. Talagrand, "Pettis integral and measure theory" ''Mem. Amer. Math. Soc.'' , '''307''' (1984)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> G.E.F. Thomas, "The Lebesgue–Nikodým theorem for vector valued Radon measures" ''Mem. Amer. Math. Soc.'' , '''139''' (1974)</TD></TR></table> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.D. Bourgin, "Geometric aspects of convex sets with the Radon–Nikodým property" , ''Lect. notes in math.'' , '''993''' , Springer (1983) {{MR|704815}} {{ZBL|0512.46017}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Dinculeanu, "Vector measures" , Pergamon (1967) {{MR|0214722}} {{MR|0206190}} {{ZBL|0992.28006}} {{ZBL|0691.60030}} {{ZBL|0647.60062}} {{ZBL|0283.60051}} {{ZBL|0195.34002}} {{ZBL|0142.10502}} {{ZBL|0178.17302}} {{ZBL|0171.01701}} {{ZBL|0117.33702}} {{ZBL|0271.28006}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Diestel, J.J. Uhl jr., "Vector measures" , ''Math. Surveys'' , '''15''' , Amer. Math. Soc. (1977) {{MR|0453964}} {{ZBL|0369.46039}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> N. Dunford, J.T. Schwartz, "Linear operators. General theory" , '''1''' , Interscience (1958) {{MR|0117523}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> I. Kluvanek, G. Knowles, "Vector measures and control systems" , North-Holland (1975) {{MR|0499068}} {{ZBL|0316.46043}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> M. Talagrand, "Pettis integral and measure theory" ''Mem. Amer. Math. Soc.'' , '''307''' (1984) {{MR|0756174}} {{ZBL|0582.46049}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> G.E.F. Thomas, "The Lebesgue–Nikodým theorem for vector valued Radon measures" ''Mem. Amer. Math. Soc.'' , '''139''' (1974) {{MR|}} {{ZBL|0282.28004}} </TD></TR></table> |
Revision as of 12:13, 27 September 2012
A finitely-additive set function defined on a field of subsets
of a set
, with values in a Banach space
(or, more generally, a topological vector space). A vector measure
is called strongly additive if
converges in
for every sequence of pairwise disjoint sets
, and countably additive if, in addition,
whenever
belongs to
. If
is countably additive for every
, then
is said to be weakly countably additive. A weakly countably-additive vector measure defined on a
-field is countably additive (the Orlicz–Pettis theorem). The variation
of
is the extended real-valued non-negative finitely-additive set function defined by
![]() |
where the supremum is over all finite partitions of
into disjoint members of
.
is said to have bounded variation if
.
is countably additive if and only if
is. The semi-variation
of
is defined by
![]() |
is a monotone finitely-subadditive set function, and if
, then
is said to have bounded semi-variation. Since, equivalently, this means that the range of
is norm bounded, such measures are also called bounded. Vector measures of bounded variation are strongly additive, and strongly-additive vector measures are bounded. A bounded vector measure is strongly additive if and only if its range is relatively weakly compact. In particular, a countably-additive vector measure has relatively weakly-compact range.
Let be a sequence of
-valued countably-additive vector measures defined on a
-field
, and let each
be
-continuous, i.e.
, where
is a non-negative extended real-valued measure. Now, if
exists for every
, then the
-continuity is uniform for
, i.e.
, uniformly in
. Hence
is
-continuous. In particular, if
is finite it follows that
is countably additive. This is the Vitali–Hahn–Saks theorem. Another striking result from the theory of vector measures is the so-called Nikodým boundedness theorem: For a collection
of bounded vector measures
on a
-field
, if
for each
, then
is uniformly bounded, i.e.
. There are also versions for strongly-additive vector measures of the well-known decomposition theorems of Yosida–Hewitt and of Lebesgue (see [a3]). Finally, a non-atomic
-valued measure on a
-field has compact and convex range if
. This is Lyapunov's theorem. It fails for infinite-dimensional
.
Vector measure theory has important applications to other areas of functional analysis. First of all to operator theory, where problems of representing operators on certain function spaces may well have been the original motive for studying vector measures. Much later, in the 1970s, the problem of differentiating vector measures led to a body of results in the geometry of Banach spaces, centering around the so-called Radon–Nikodým property. Below these developments are given briefly (see also [a1] and [a4]); see [a5] for the role of vector measures in control theory.
Let be a compact Hausdorff space,
the space of continuous functions on
with the sup-norm, and
a bounded linear operator (
is any Banach space). Then
can be represented by a weak-
countably-additive vector measure
defined on the
-field of Borel sets in
and taking its values in
, the bidual of
(cf. Adjoint space). This representation is particularly satisfactory when
is weakly compact, for then
has its values in
, and is countably additive (either of these properties is in fact equivalent to
being weakly compact). Then one has
(
), where the integral has its more or less obvious meaning. An immediate consequence of this representation formula is that
maps weakly-compact sets into norm-compact sets (
has the Dunford–Pettis property). Other classes of operators
such as the compact, the nuclear and the absolutely summing ones admit equally nice characterizations in terms of their representing measures (see [a3]).
Now, let be a bounded linear operator from
into a Banach space
(
a finite measure space). There is an obvious vector measure
associated to
:
,
. Moreover,
is
-continuous and of bounded variation. If
has a Radon–Nikodým derivative, i.e. if there exists an
-valued Bochner-integrable function
on
such that
(
), then
can be represented as a Bochner integral:
(
). It is known, however, that in general such a derivative
does not exist. If, for a particular
and for any measure space
, every
-continuous
-valued measure of bounded variation has a Radon–Nikodým derivative, then
is said to have the Radon–Nikodým property (RNP). Examples of spaces with the RNP: separable dual spaces (the Dunford–Pettis theorem) and reflexive spaces, so in particular Hilbert spaces. The spaces
(i.e. the space of null sequences) and
fail the RNP. The RNP for
has been shown to be equivalent to various convergence properties for
-valued martingales. In turn, this martingale approach has led to various purely geometrical characterizations of spaces with the RNP (see [a1] for details). An example is as follows:
has the RNP if and only if for every closed bounded convex subset
and every
there is a closed hyperplane
in
so that both half-spaces determined by
intersect
, and one of these intersections has diameter
(
is dentable). The Krein–Milman property states that every closed bounded convex set of
is the norm-closed hull of its extreme points. If a Banach space possesses the RNP, then it has the Krein–Milman property (J. Lindenstrauss). For dual spaces
these two properties are equivalent.
The question can also be asked which -continuous
-valued measures are Pettis integrals (rather than Bochner integrals, cf. Pettis integral). This leads to the so-called weak Radon–Nikodým property (WRNP) (see [a6]).
References
[a1] | R.D. Bourgin, "Geometric aspects of convex sets with the Radon–Nikodým property" , Lect. notes in math. , 993 , Springer (1983) MR704815 Zbl 0512.46017 |
[a2] | N. Dinculeanu, "Vector measures" , Pergamon (1967) MR0214722 MR0206190 Zbl 0992.28006 Zbl 0691.60030 Zbl 0647.60062 Zbl 0283.60051 Zbl 0195.34002 Zbl 0142.10502 Zbl 0178.17302 Zbl 0171.01701 Zbl 0117.33702 Zbl 0271.28006 |
[a3] | J. Diestel, J.J. Uhl jr., "Vector measures" , Math. Surveys , 15 , Amer. Math. Soc. (1977) MR0453964 Zbl 0369.46039 |
[a4] | N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523 |
[a5] | I. Kluvanek, G. Knowles, "Vector measures and control systems" , North-Holland (1975) MR0499068 Zbl 0316.46043 |
[a6] | M. Talagrand, "Pettis integral and measure theory" Mem. Amer. Math. Soc. , 307 (1984) MR0756174 Zbl 0582.46049 |
[a7] | G.E.F. Thomas, "The Lebesgue–Nikodým theorem for vector valued Radon measures" Mem. Amer. Math. Soc. , 139 (1974) Zbl 0282.28004 |
Vector measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_measure&oldid=17378