Local measurability

In mathematics , more precisely in measure theory , local measurability is a property that functions.

definition

Be a measurement space and a measurement space . An illustration is locally measurable if for each of the image is measured, d. H. if for each is always . ${\displaystyle (\Omega ,{\mathcal {A}},\mu )}$ ${\displaystyle (S,{\mathcal {B}})}$ ${\displaystyle f\colon \Omega \to S}$ ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle \mu (A)<\infty }$ ${\displaystyle f|_{A}\colon (A,{\mathcal {A}}\cap A)\to (S,{\mathcal {B}})}$ ${\displaystyle B\in {\mathcal {B}}}$ ${\displaystyle f^{-1}(B)\cap A\in {\mathcal {A}}}$

properties

• Every measurable function can also be measured locally.
• If a σ-finite measure space , then every locally measurable function is also measurable, but in general this is wrong. ${\displaystyle (\Omega ,{\mathcal {A}},\mu )}$

literature

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