# vector measure

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

A vector measure is a function $\mu:\mathcal{F}\longrightarrow X$ that is , i.e. for any two disjoint sets $A_{1},A_{2}$ in $\mathcal{F}$ 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 $\mathcal{F}$ such that $\displaystyle\bigcup_{n=1}^{\infty}A_{n}\in\mathcal{F}$ one has

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

where the series converges in the topology of $X$.

In the particular case when $X=\mathbb{C}$, 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 $(X,\mathfrak{B},\lambda)$ be a measure space. Consider the Banach space $L^{p}(X,\mathfrak{B},\lambda)$ (http://planetmath.org/LpSpace) with $1\leq p\leq\infty$. Define the the function $\mu:\mathfrak{B}\longrightarrow L^{p}(X,\mathfrak{B},\mu)$ 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 $1\leq p<\infty$ (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.

 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