Vector measure

A finitely-additive set function $ F $ defined on a field of subsets $ {\mathcal F} $ of a set $ \Omega $, with values in a Banach space $ X $( or, more generally, a topological vector space). A vector measure $ F $ is called strongly additive if $ \sum _ {n=} 1 ^ \infty F( E _ {n} ) $ converges in $ X $ for every sequence of pairwise disjoint sets $ E _ {n} \in {\mathcal F} $, and countably additive if, in addition, $ \sum _ {n=} 1 ^ \infty F( E _ {n} ) = F ( \cup _ {n=} 1 ^ \infty E _ {n} ) $ whenever $ \cup _ {n=} 1 ^ \infty E _ {n} $ belongs to $ {\mathcal F} $. If $ x ^ {*} F $ is countably additive for every $ x ^ {*} \in X ^ {*} $, then $ F $ is said to be weakly countably additive. A weakly countably-additive vector measure defined on a $ \sigma $- field is countably additive (the Orlicz–Pettis theorem). The variation $ | F | $ of $ F $ is the extended real-valued non-negative finitely-additive set function defined by

$$ | F | ( E) = \sup _ \pi \ \sum _ {A \in \pi } \| F( A) \| ,\ \ E \in {\mathcal F} , $$

where the supremum is over all finite partitions $ \pi $ of $ E $ into disjoint members of $ {\mathcal F} $. $ F $ is said to have bounded variation if $ | F | ( \Omega ) < \infty $. $ | F | $ is countably additive if and only if $ F $ is. The semi-variation $ \| F \| $ of $ F $ is defined by

$$ \| F \| ( E) = \sup \{ {| x ^ {*} F | ( E) } : { \| x ^ {*} \| \leq 1 } \} ,\ \ E \in {\mathcal F} . $$

$ \| F \| $ is a monotone finitely-subadditive set function, and if $ \| F \| ( \Omega ) < \infty $, then $ F $ is said to have bounded semi-variation. Since, equivalently, this means that the range of $ F $ 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 $ ( F _ {n} ) $ be a sequence of $ X $- valued countably-additive vector measures defined on a $ \sigma $- field $ \Sigma $, and let each $ F _ {n} $ be $ \mu $- continuous, i.e. $ \lim\limits _ {\mu ( E) \rightarrow 0 } F _ {n} ( E) = 0 $, where $ \mu $ is a non-negative extended real-valued measure. Now, if $ \lim\limits _ {n \rightarrow \infty } F _ {n} ( E) = F( E) $ exists for every $ E \in \Sigma $, then the $ \mu $- continuity is uniform for $ n \in \mathbf N $, i.e. $ \lim\limits _ {\mu ( E) \rightarrow 0 } F _ {n} ( E) = 0 $, uniformly in $ n $. Hence $ F $ is $ \mu $- continuous. In particular, if $ \mu $ is finite it follows that $ F $ 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 $ M $ of bounded vector measures $ F $ on a $ \sigma $- field $ \Sigma $, if $ \sup _ {F \in M } \| F( E) \| < \infty $ for each $ E \in \Sigma $, then $ M $ is uniformly bounded, i.e. $ \sup _ {F \in M , E \in \Sigma } \| F( E) \| < \infty $. 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 $ X $- valued measure on a $ \sigma $- field has compact and convex range if $ \mathop{\rm dim} X < \infty $. This is Lyapunov's theorem. It fails for infinite-dimensional $ X $.

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 $ \Omega $ be a compact Hausdorff space, $ C( \Omega ) $ the space of continuous functions on $ \Omega $ with the sup-norm, and $ T : C( \Omega ) \rightarrow X $ a bounded linear operator ( $ X $ is any Banach space). Then $ T $ can be represented by a weak- $ * $ countably-additive vector measure $ F $ defined on the $ \sigma $- field of Borel sets in $ \Omega $ and taking its values in $ X ^ {**} $, the bidual of $ X $( cf. Adjoint space). This representation is particularly satisfactory when $ T $ is weakly compact, for then $ F $ has its values in $ X $, and is countably additive (either of these properties is in fact equivalent to $ T $ being weakly compact). Then one has $ T f = \int _ \Omega f dF $( $ f \in C( \Omega ) $), where the integral has its more or less obvious meaning. An immediate consequence of this representation formula is that $ T $ maps weakly-compact sets into norm-compact sets ( $ C( \Omega ) $ has the Dunford–Pettis property). Other classes of operators $ T : C( \Omega ) \rightarrow X $ 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 $ T $ be a bounded linear operator from $ L _ {1} ( \Omega , \Sigma , \mu ) $ into a Banach space $ X $( $ ( \Omega , \Sigma , \mu ) $ a finite measure space). There is an obvious vector measure $ F $ associated to $ T $: $ F( E) = T ( \chi _ {E} ) $, $ E \in \Sigma $. Moreover, $ F $ is $ \mu $- continuous and of bounded variation. If $ F $ has a Radon–Nikodým derivative, i.e. if there exists an $ X $- valued Bochner-integrable function $ f $ on $ \Omega $ such that $ F( E) = \int _ {E} f d \mu $( $ E \in \Sigma $), then $ T $ can be represented as a Bochner integral: $ Tg = \int _ \Omega g f d \mu $( $ g \in L _ {1} ( \mu ) $). It is known, however, that in general such a derivative $ f $ does not exist. If, for a particular $ X $ and for any measure space $ ( \Omega , \Sigma , \mu ) $, every $ \mu $- continuous $ X $- valued measure of bounded variation has a Radon–Nikodým derivative, then $ X $ 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 $ c _ {0} $( i.e. the space of null sequences) and $ L _ {1} [ 0, 1] $ fail the RNP. The RNP for $ X $ has been shown to be equivalent to various convergence properties for $ X $- 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: $ X $ has the RNP if and only if for every closed bounded convex subset $ B \subset X $ and every $ \epsilon > 0 $ there is a closed hyperplane $ H $ in $ X $ so that both half-spaces determined by $ H $ intersect $ B $, and one of these intersections has diameter $ < \epsilon $( $ X $ is dentable). The Krein–Milman property states that every closed bounded convex set of $ X $ 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 $ X ^ {*} $ these two properties are equivalent.

The question can also be asked which $ \mu $- continuous $ X $- 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