# 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.

## 0.0.1 Examples :

 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