Vector measure

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

1 Definitions and first consequences
2 Examples
3 The variation of a vector measure
4 References

[edit] 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 $\mu:\mathcal {F} \to X$ such that for any two disjoint sets $A$ and $B$ in $\mathcal{F}$ one has

$\mu(A\cup B) =\mu(A) + \mu (B).$

A vector measure $μ$ 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

$\mu\left(\displaystyle\bigcup_{i=1}^\infty A_i\right) =\sum_{i=1}^{\infty}\mu(A_i)$

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 $μ$ is countably additive if and only if for any sequence $(A_i)_{i=1, 2, \dots}$ as above one has

$\lim_{n\to\infty}\left\|\mu\left(\displaystyle\bigcup_{i=n}^\infty A_i\right)\right\|=0, \quad\quad\quad (*)$

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.

[edit] 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 $χ$ is the indicator function of $A .$ Depending on where $μ$ is declared to take values, we get two different outcomes.

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

$μ,$ 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.

[edit] The variation of a vector measure

Given a vector measure $\mu:\mathcal{F}\to X,$ the variation $| μ |$ of $μ$ is defined as

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

where the supremum is taken over all the partitions

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

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 $μ$ 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 $| μ | (Ω)$ is finite, the measure $μ$ is said to be of bounded variation. One can prove that if $μ$ is a vector measure of bounded variation, then $μ$ is countably additive if and only if $| μ |$ is countably additive.