vector measure
Let $S$ be a set and $\mathcal{F}$ a field of sets of $S$. Let $X$ be a topological vector space^{}.
A vector measure is a function $\mu :\mathcal{F}\u27f6X$ that is , i.e. for any two disjoint sets ${A}_{1},{A}_{2}$ in $\mathcal{F}$ we have
$$\mu ({A}_{1}\cup {A}_{2})=\mu ({A}_{1})+\mu ({A}_{2})$$ 
A vector measure $\mu $ is said to be if for any sequence ${({A}_{n})}_{n\in \mathbb{N}}$ of disjoint sets in $\mathcal{F}$ such that $\bigcup _{n=1}^{\mathrm{\infty}}}{A}_{n}\in \mathcal{F$ one has
$$\mu (\bigcup _{n=1}^{\mathrm{\infty}}{A}_{n})=\sum _{n=1}^{\mathrm{\infty}}\mu ({A}_{n})$$ 
where the series converges in the topology of $X$.
In the particular case when $X=\u2102$, a countably additive vector measure is usually called a complex measure.
Thus, vector measures are to measures^{} and signed measures but they take values on a vector space (with a particular topology).
0.0.1 Examples :

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Let $(X,\U0001d505,\lambda )$ be a measure space. Consider the Banach space^{} ${L}^{p}(X,\U0001d505,\lambda )$ (http://planetmath.org/LpSpace) with $1\le p\le \mathrm{\infty}$. Define the the function $\mu :\U0001d505\u27f6{L}^{p}(X,\U0001d505,\mu )$ by
$$\mu (A):={\chi}_{A}$$ where ${\chi}_{A}$ denotes the characteristic function^{} of the measurable set^{} $A$. It is easily seen that $\mu $ is a vector measure, which is countably additive if $$ (in case $p=\mathrm{\infty}$, countably additiveness fails).

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spectral measures are vector measures in the $\sigma $algebra of Borel sets in $\u2102$ whose values are projections on some Hilbert space^{}. They are used in general formulations of the spectral theorem^{}.
Title  vector measure 
Canonical name  VectorMeasure 
Date of creation  20130322 17:29:23 
Last modified on  20130322 17:29:23 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  12 
Author  asteroid (17536) 
Entry type  Definition 
Classification  msc 47A56 
Classification  msc 46G12 
Classification  msc 46G10 
Classification  msc 28C20 
Classification  msc 28B05 
Defines  complex measure 
Defines  countably additive vector measure 