Vector measure

In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties.

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 $mu:mathcal\; \{F\}\; o\; X$ such that for any two disjoint sets $A$ and $B$ in $mathcal\{F\}$ one has

: $mu(Acup\; B)\; =mu(A)\; +\; mu\; (B).$

A vector measure $mu$ is called countably additive if for any sequence $(A\_i)\_\{i=1,\; 2,\; dots\}$ of disjoint sets in $mathcal\; F$ such that their union is in $mathcal\; F$ it holds that

:

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 $(A\_i)\_\{i=1,\; 2,\; dots\}$ as above one has

:

where $|cdot|$ is the norm on $X.$

Countably additive vector measures defined on sigma-algebras are more general than measures, signed measures, and complex measures, which are countably additive functions taking values respectively on the extended interval $[0,\; infty]\; ,$ 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, we get two different outcomes.

* $mu,$ viewed as a function from $mathcal\; F$ to the Lp space $L^infty(\; [0,\; 1]\; ),$ is a vector measure which is not countably-additive.

* $mu,$ viewed as a function from $mathcal\; F$ to the Lp space $L^1(\; [0,\; 1]\; ),$ 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 $mu:mathcal\{F\}\; o\; X,$ the variation $|mu|$ of $mu$ is defined as

: $|mu|(A)=sup\; sum\_\{i=1\}^n\; |mu(A\_i)|$

where the supremum is taken over all the partitions

:

of $A$ into a finite number of disjoint sets, for all $A$ in $mathcal\{F\}$. Here, $|cdot|$ is the norm on $X.$

The variation of $mu$ is a finitely additive function taking values in $[0,\; infty]\; .$ It holds that

: $||mu(A)||le\; |mu|(A)$

for any $A$ in $mathcal\{F\}.$ 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.

References

*cite book
last = Diestel
first = J.
coauthors = Uhl, Jr., J. J.
title = Vector measures
publisher = Providence, R.I: American Mathematical Society
date = 1977
pages =
isbn = 0821815156
*