Locatable dimension space

In mathematics , more precisely in measure theory , localizability is a property that belongs to a measure space .

definition

A measurement space is called localizable if the following applies: Is and a family of measurable functions with for all with so there is a locally measurable function with for all . ${\displaystyle (S,{\mathcal {A}},\mu )}$ ${\displaystyle {\mathcal {A}}_{0}:=\{A\in {\mathcal {A}}:\mu (A)<\infty \}}$ ${\displaystyle (g_{A})_{A\in {\mathcal {A}}_{0}}}$ ${\displaystyle g_{A}\colon A\to \mathbb {R} }$ ${\displaystyle g_{A}|_{A\cap B}=g_{B}|_{A\cap B}}$ ${\displaystyle A,B\in {\mathcal {A}}}$ ${\displaystyle \mu (A),\mu (B)<\infty }$ ${\displaystyle g\colon S\to \mathbb {R} }$ ${\displaystyle g|_{A}=g_{A}}$ ${\displaystyle A\in {\mathcal {A}}_{0}}$

Explanation

In a localizable measurement space , it is therefore possible to combine locally consistently given measurable functions into a (locally) measurable function that is defined over the entire space. Local here means to sets of finite measure.

properties

• Perhaps the most important property of a localizable measurement space is that in localizable spaces the dual space of can be described as the space of locally measurable, locally essentially limited functions. In the case of σ-finite measure spaces, this space coincides with the usual one. ${\displaystyle L_{1}}$ ${\displaystyle L_{\infty }}$

literature

• Ehrhard Behrends: Measure and integration theory. Springer, Berlin et al. 1987, ISBN 3-540-17850-3 , Section IV.3, pp. 184-192.