vector measure
Let be a set and a field of sets of . Let be a topological vector space.
A vector measure is a function that is , i.e. for any two disjoint sets in we have
A vector measure is said to be if for any sequence of disjoint sets in such that one has
where the series converges in the topology of .
In the particular case when , 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 be a measure space. Consider the Banach space (http://planetmath.org/LpSpace) with . Define the the function by
where denotes the characteristic function of the measurable set . It is easily seen that is a vector measure, which is countably additive if (in case , countably additiveness fails).
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spectral measures are vector measures in the -algebra of Borel sets in 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 | 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 |