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 .
![(\ Omega, \ mathcal A, \ mu)](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cabc7dd1ffb04becc182e5356682ec027db7211)
![{\ displaystyle f \ colon \ Omega \ to S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bc41010aa82e7d1d7177087d6b1178957811ded)
![A \ in {\ mathcal A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6e69647797be244cf2ebc28ecd61fafba8790c1)
![\ mu (A) <\ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a97dfa93ce05b5c7fbac7bec0f21dfd8cbde288)
![{\ displaystyle f | _ {A} \ colon (A, {\ mathcal {A}} \ cap A) \ to (S, {\ mathcal {B}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d20f91607d97dee70e50156264d4bf0e707f48c)
![B \ in {\ mathcal B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0df7ead8ec7e88c3e04464929ae1213bbc1cd13)
![f ^ {{- 1}} (B) \ cap A \ in {\ mathcal A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a7b872c9cad969b5b76d1b1d4c8db84a73b7c6b)
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.
![(\ Omega, {\ mathcal A}, \ mu)](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cabc7dd1ffb04becc182e5356682ec027db7211)
literature
- Ehrhard Behrends: Measure and integration theory. Springer, Berlin et al. 1987, ISBN 3-540-17850-3 , Section IV.3, pp. 184-192.