Definitions

Vector_measure

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} to 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

muleft(displaystylebigcup_{i=1}^infty A_iright) =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_{ntoinfty}left|muleft(displaystylebigcup_{i=n}^infty A_iright)right|=0, quadquadquad (*)

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}to 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=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 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

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

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