Vitali–Carathéodory theorem
Jump to navigation
Jump to search
This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages)
|
In mathematics, the Vitali–Carathéodory theorem is a result in real analysis that shows that, under the conditions stated below, integrable functions can be approximated in L1 from above and below by lower- and upper-semicontinuous functions, respectively. It is named after Giuseppe Vitali and Constantin Carathéodory.
Statement of the theorem
Let X be a locally compact Hausdorff space equipped with a Borel measure, μ, that is finite on every compact set, outer regular, and tight when restricted to any Borel set that is open or of finite mass. If f is an element of L1(μ) then, for every ε > 0, there are functions u and v on X such that u ≤ f ≤ v, u is upper-semicontinuous and bounded above, v is lower-semicontinuous and bounded below, and
References
- Rudin, Walter (1986). Real and Complex Analysis (third ed.). McGraw-Hill. pp. 56–57. ISBN 978-0-07-054234-1.