Definitions

# 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 $\left(Omega, mathcal F\right)$ and a Banach space $X$, a finitely additive vector measure (or measure, for short) is a function $mu:mathcal \left\{F\right\} to X$ such that for any two disjoint sets $A$ and $B$ in $mathcal\left\{F\right\}$ one has

$mu\left(Acup B\right) =mu\left(A\right) + mu \left(B\right).$

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

$muleft\left(displaystylebigcup_\left\{i=1\right\}^infty A_iright\right) =sum_\left\{i=1\right\}^\left\{infty\right\}mu\left(A_i\right)$

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 $\left(A_i\right)_\left\{i=1, 2, dots\right\}$ as above one has

$lim_\left\{ntoinfty\right\}left|muleft\left(displaystylebigcup_\left\{i=n\right\}^infty A_iright\right)right|=0, quadquadquad \left(*\right)$

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 $\left[0, infty\right],$ the set of real numbers, and the set of complex numbers.

## Examples

Consider the field of sets made up of the interval $\left[0, 1\right]$ together with the family $mathcal F$ of all Lebesgue measurable sets contained in this interval. For any such set $A$, define

$mu\left(A\right)=chi_A,$

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 $L^infty\left(\left[0, 1\right]\right),$ is a vector measure which is not countably-additive.
• $mu,$ viewed as a function from $mathcal F$ to the Lp space $L^1\left(\left[0, 1\right]\right),$ 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 $mu:mathcal\left\{F\right\}to X,$ the variation $|mu|$ of $mu$ is defined as

$|mu|\left(A\right)=sup sum_\left\{i=1\right\}^n |mu\left(A_i\right)|$

where the supremum is taken over all the partitions

$A=bigcup_\left\{i=1\right\}^n A_i$

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

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

$||mu\left(A\right)||le |mu|\left(A\right)$

for any $A$ in $mathcal\left\{F\right\}.$ If $|mu|\left(Omega\right)$ 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

• Diestel, J.; Uhl, Jr., J. J. Vector measures. Providence, R.I: American Mathematical Society.

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