, 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
and a Banach space
, a finitely additive vector measure
, for short) is a function
such that for any two disjoint sets
A vector measure is called countably additive if for any sequence of disjoint sets in such that their union is in it holds that
with the series on the right-hand side convergent in the norm of the Banach space
It can be proved that an additive vector measure is countably additive if and only if for any sequence as above one has
where is the norm on
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 the set of real numbers, and the set of complex numbers.
Consider the field of sets made up of the interval together with the family of all Lebesgue measurable sets contained in this interval. For any such set , define
where is the indicator function of Depending on where is declared to take values, we get two different outcomes.
- viewed as a function from to the Lp space is a vector measure which is not countably-additive.
- viewed as a function from to the Lp space 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 the variation of is defined as
where the supremum is taken over all the partitions
of into a finite number of disjoint sets, for all in . Here, is the norm on
The variation of is a finitely additive function taking values in It holds that
for any in If is finite, the measure is said to be of bounded variation. One can prove that if is a vector measure of bounded variation, then is countably additive if and only if is countably additive.
- Diestel, J.; Uhl, Jr., J. J. Vector measures. Providence, R.I: American Mathematical Society.