In mathematics, a vector measure is a function defined on a

family of sets In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set fa ...

and taking

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

values satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real values only.

Definitions and first consequences

Given a field of sets

(\Omega, \mathcal F)

and a

Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...

X,

a finitely additive vector measure (or measure, for short) is a function

\mu:\mathcal  \to X

such that for any two disjoint sets

A

and

B

\mathcal

one has

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

A vector measure

\mu

is called countably additive if for any

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...

(A_i)_^

of disjoint sets in

\mathcal F

such that their union is in

\mathcal F

it holds that

\mu = \sum_^\mu(A_i)

with the series on the right-hand side convergent in the norm of the Banach space

X.