- Vector measure
In

mathematics , a**vector measure**is a function defined on afamily of sets and taking vector values satisfying certain properties.**Definitions and first consequences**Given a

field of sets $(Omega,\; mathcal\; F)$ and aBanach space $X$, a**finitely additive vector measure**(or**measure**, for short) is a function $mu:mathcal\; \{F\}\; o\; X$ such that for any twodisjoint set s $A$ and $B$ in $mathcal\{F\}$ one has: $mu(Acup\; B)\; =mu(A)\; +\; mu\; (B).$

A vector measure $mu$ is called

**countably additive**if for anysequence $(A\_i)\_\{i=1,\; 2,\; dots\}$ of disjoint sets in $mathcal\; F$ such that their union is in $mathcal\; F$ it holds that: $muleft(displaystyleigcup\_\{i=1\}^infty\; A\_i\; ight)\; =sum\_\{i=1\}^\{infty\}mu(A\_i)$

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

: $lim\_\{n\; oinfty\}left|muleft(displaystyleigcup\_\{i=n\}^infty\; A\_i\; ight)\; ight|=0,\; quadquadquad\; (*)$

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

Countably additive vector measures defined on

sigma-algebra s are more general than measures,signed measure s, andcomplex measure s, which arecountably additive function s taking values respectively on the extended interval $[0,\; infty]\; ,$ the set ofreal number s, and the set ofcomplex number s.**Examples**Consider the field of sets made up of the interval $[0,\; 1]$ together with the family $mathcal\; F$ of all

Lebesgue measurable set s 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: $A=igcup\_\{i=1\}^n\; A\_i$

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

*

*Wikimedia Foundation.
2010.*

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