Lyapunov Vector-measure Theorem

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.

Definitions and first consequences

Given a field of sets $(\Omega, \mathcal F)$ and a Banach space $X,$ a finitely additive vector measure (or measure, for short) is a function $\mu:\mathcal \to X$ such that for any two disjoint sets $A$ and $B$ in $\mathcal$ one has
$\mu(A\cup B) =\mu(A) + \mu (B).$

A vector measure $\mu$ is called countably additive if for any sequence $(A_i)_^$ of disjoint sets in $\mathcal F$ such that their union is in $\mathcal F$ it holds that
$\mu = \sum_^\mu(A_i)$
with the series on the right-hand side convergent in the norm of the Banach space $X.$

It can be proved that an additive vector measure $\mu$ is countably additive if and only if for any sequence $(A_i)_^$ as above one has

where $\, \cdot\,$ is the norm on $X.$

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 , \infty), the set of real numbers, and the set of complex number">real_number.html" ;"title=", \infty), the set of real number">, \infty), the set of real numbers, and the set of complex numbers.

Examples

Consider the field of sets made up of the interval $[0, 1]$ together with the family $\mathcal F$ of all Lebesgue measurable sets contained in this interval. For any such set $A,$ define
$\mu(A) = \chi_A$
where $\chi$ is the indicator function of $A.$ Depending on where $\mu$ is declared to take values, two different outcomes are observed.

* $\mu,$ viewed as a function from $\mathcal F$ to the $L^p$-space L^\infty( , 1, is a vector measure which is not countably-additive.
* $\mu,$ viewed as a function from $\mathcal F$ to the $L^p$-space L^1( , 1, is a countably-additive vector measure.

Both of these statements follow quite easily from the criterion () stated above.

The variation of a vector measure

Given a vector measure $\mu : \mathcal \to X,$ the variation $, \mu,$ of $\mu$ is defined as
$, \mu, (A)=\sup \sum_^n \, \mu(A_i)\,$
where the supremum is taken over all the partitions
$A = \bigcup_^n A_i$
of $A$ into a finite number of disjoint sets, for all $A$ in $\mathcal.$ Here, $\, \cdot\,$ is the norm on $X.$

The variation of $\mu$ is a finitely additive function taking values in , \infty It holds that
$\, \mu(A)\, \leq , \mu, (A)$
for any $A$ in $\mathcal.$ If $, \mu, (\Omega)$ is finite, the measure $\mu$ is said to be of bounded variation. One can prove that if $\mu$ is a vector measure of bounded variation, then $\mu$ is countably additive if and only if $, \mu,$ is countably additive.

Lyapunov's theorem

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. Kluvánek, I., Knowles, G., ''Vector Measures and Control Systems'', North-Holland Mathematics Studies 20, Amsterdam, 1976. 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). It is used in economics, This paper builds on two papers by Aumann:

in ( "bang–bang") control theory, and in statistical theory.
Lyapunov's theorem has been proved by using the Shapley–Folkman lemma, which has been viewed as a discrete analogue of Lyapunov's theorem.Page 210:

See also

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References

Bibliography

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* Kluvánek, I., Knowles, G, ''Vector Measures and Control Systems'', North-Holland Mathematics Studies 20, Amsterdam, 1976.
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{{Measure theory

Control theory
Functional analysis
Measures (measure theory)