Vector measure

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.

Given a field of sets ${\displaystyle (\Omega ,{\mathcal {F}})}$ and a Banach space ${\displaystyle X}$, a finitely additive vector measure (or measure, for short) is a function ${\displaystyle \mu :{\mathcal {F}}\to X}$ such that for any two disjoint sets ${\displaystyle A}$ and ${\displaystyle B}$ in ${\displaystyle {\mathcal {F}}}$ one has

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