vector measure
Let S be a set and ℱ a field of sets of S. Let X be a topological vector space.
A vector measure is a function μ:ℱ⟶X that is , i.e. for any two disjoint sets A1,A2 in ℱ we have
μ(A1∪A2)=μ(A1)+μ(A2) |
A vector measure μ is said to be if for any sequence (An)n∈ℕ of disjoint sets in ℱ such that ∞⋃n=1An∈ℱ one has
μ(∞⋃n=1An)=∞∑n=1μ(An) |
where the series converges in the topology of X.
In the particular case when X=ℂ, 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,𝔅,λ) be a measure space. Consider the Banach space
Lp(X,𝔅,λ) (http://planetmath.org/LpSpace) with 1≤p≤∞. Define the the function μ:𝔅⟶Lp(X,𝔅,μ) by
μ(A):= 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 |