Vector measure explained

In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real values only.

Definitions and first consequences

(\Omega, l F) and a Banach space X, a finitely additive vector measure (or measure, for short) is a function \mu: l {F} \to X such that for any two disjoint sets A and B in l{F} one has$$\backslash mu(A\backslash cup\; B)\; =\backslash mu(A)\; +\; \backslash mu\; (B).$$

A vector measure

\mu is called countably additive if for any sequence (A _{i)} ^{inf ty} ^{i =1} of disjoint sets in l F such that their union is in l F it holds that$$\backslash mu\; =\; \backslash sum\_^\backslash mu(A\_i)$$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)} ^{inf ty} ^{i =1} as above one has