PlanetMath: vector measure

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vector measure

(Definition)

Let be a set and $\mathcal{F}$ a field of sets of . Let be a topological vector space.

A vector measure is a function $\mu : \mathcal{F} \longrightarrow X$ that is finitely additive, i.e. for any two disjoint sets in $\mathcal{F}$ we have

$\displaystyle \mu(A_1 \cup A_2) = \mu(A_1)+\mu(A_2)$

A vector measure $\mu$ is said to be countably additive if for any sequence $(A_n)_{n \in \mathbb{N}}$ of disjoint sets in $\mathcal{F}$ such that $\displaystyle \bigcup_{n =1}^{\infty}A_n \in \mathcal{F}$ one has

$\displaystyle \mu(\bigcup_{n =1}^{\infty}A_n) = \sum_{n=1}^{\infty} \mu(A_n)$

where the series converges in the topology of

In the particular case when $X = \mathbb{C}$ , a countably additive vector measure is usually called a complex measure.

Thus, vector measures are similar to measures and signed measures but they take values on a vector space (with a particular topology).

Examples :

Let $(X, \mathfrak{B}, \lambda)$ be a measure space. Consider the Banach space $L^p(X, \mathfrak{B}, \lambda)$ with $1 \leq p \leq \infty$ . Define the the function $\mu : \mathfrak{B} \longrightarrow L^p(X, \mathfrak{B}, \mu)$ by
$\displaystyle \mu(A) := \chi_A$
where $\chi_A$ denotes the characteristic function of the measurable set . It is easily seen that $\mu$ is a vector measure, which is countably additive if $1 \leq p < \infty$ (in case $p = \infty$ , countably additiveness fails).

spectral measures are vector measures in the $\sigma$ -algebra of Borel sets in $\mathbb{C}$ whose values are projections on some Hilbert space. They are used in general formulations of the spectral theorem.

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Also defines:

complex measure, countably additive vector measure

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Cross-references: spectral theorem, Hilbert space, projections, Borel sets, spectral measures, countably additive, measurable set, characteristic function, Banach space, measure space, vector space, signed measures, measures, topology, converges, series, sequence, disjoint, function, topological vector space, field of sets
There are 4 references to this entry.

This is version 9 of vector measure, born on 2007-08-19, modified 2008-03-13.
Object id is 9877, canonical name is VectorMeasure.
Accessed 1314 times total.

Classification:

AMS MSC:	28B05 (Measure and integration :: Set functions, measures and integrals with values in abstract spaces :: Vector-valued set functions, measures and integrals)
	28C20 (Measure and integration :: Set functions and measures on spaces with additional structure :: Set functions and measures and integrals in infinite-dimensional spaces )
	46G10 (Functional analysis :: Measures, integration, derivative, holomorphy :: Vector-valued measures and integration)
	46G12 (Functional analysis :: Measures, integration, derivative, holomorphy :: Measures and integration on abstract linear spaces)
	47A56 (Operator theory :: General theory of linear operators :: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones)

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