In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties.
It is a generalization of the concept of finite measure, which takes nonnegative real values only.
a finitely additive vector measure (or measure, for short) is a function
is called countably additive if for any sequence
with the series on the right-hand side convergent in the norm of the Banach space
It can be proved that an additive vector measure
is countably additive if and only if for any sequence
Countably additive vector measures defined on sigma-algebras are more general than finite measures, finite signed measures, and complex measures, which are countably additive functions taking values respectively on the real interval
Consider the field of sets made up of the interval
of all Lebesgue measurable sets contained in this interval.
is declared to take values, two different outcomes are observed.
Both of these statements follow quite easily from the criterion (*) stated above.
into a finite number of disjoint sets, for all
is a finitely additive function taking values in
is a vector measure of bounded variation, then
In the theory of vector measures, Lyapunov's theorem states that the range of a (non-atomic) finite-dimensional vector measure is closed and convex.
[1][2][3] In fact, the range of a non-atomic vector measure is a zonoid (the closed and convex set that is the limit of a convergent sequence of zonotopes).
[8] Lyapunov's theorem has been proved by using the Shapley–Folkman lemma,[9] which has been viewed as a discrete analogue of Lyapunov's theorem.
"Markets with a continuum of traders".
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MR 0185073.The concept of a convex set (i.e., a set containing the segment connecting any two of its points) had repeatedly been placed at the center of economic theory before 1964.
It appeared in a new light with the introduction of integration theory in the study of economic competition: If one associates with every agent of an economy an arbitrary set in the commodity space and if one averages those individual sets over a collection of insignificant agents, then the resulting set is necessarily convex.
[Debreu appends this footnote: "On this direct consequence of a theorem of A. A.
But explanations of the ... functions of prices ... can be made to rest on the convexity of sets derived by that averaging process.
Convexity in the commodity space obtained by aggregation over a collection of insignificant agents is an insight that economic theory owes ... to integration theory.
[Italics added] Debreu, Gérard (March 1991).
"The Mathematization of economic theory".
The American Economic Review.
Presidential address delivered at the 103rd meeting of the American Economic Association, 29 December 1990, Washington, DC.