vector measure

Let S be a set and a field of sets of S. Let X be a topological vector spaceMathworldPlanetmath.

A vector measure is a function μ:X that is , i.e. for any two disjoint sets A1,A2 in we have


A vector measure μ is said to be if for any sequence (An)n of disjoint sets in such that n=1An one has


where the series converges in the topology of X.

In the particular case when X=, a countably additive vector measure is usually called a complex measure.

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

0.0.1 Examples :

  • Let (X,𝔅,λ) be a measure space. Consider the Banach spaceMathworldPlanetmath Lp(X,𝔅,λ) ( with 1p. Define the the function μ:𝔅Lp(X,𝔅,μ) by


    where χA denotes the characteristic functionMathworldPlanetmathPlanetmathPlanetmath of the measurable setMathworldPlanetmath A. It is easily seen that μ is a vector measure, which is countably additive if 1p< (in case p=, countably additiveness fails).

Title vector measure
Canonical name VectorMeasure
Date of creation 2013-03-22 17:29:23
Last modified on 2013-03-22 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