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 $\mu :ℱ⟶X$ that is , i.e. for any two disjoint sets $A_1,A_2$ in $ℱ$ 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 $ℱ$ such that ${\displaystyle \bigcup _{n=1}^{\mathrm{\infty }}}{A}_n\in ℱ$ 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 $X=ℂ$, 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 :

• Let $(X,𝔅,\lambda )$ be a measure space. Consider the Banach space $L^p(X,𝔅,\lambda )$ (http://planetmath.org/LpSpace) with $1\le p\le \mathrm{\infty }$. Define the the function $\mu :𝔅⟶L^p(X,𝔅,\mu )$ 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).

• spectral measures are vector measures in the $\sigma$-algebra of Borel sets in $ℂ$ 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