Vitali–Carathéodory theorem

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

[math]\displaystyle{ \int_X (v - u) \,\mathrm{d}\mu \lt \varepsilon. }[/math]

References

• Rudin, Walter (1986). Real and Complex Analysis (third ed.). McGraw-Hill. pp. 56–57. ISBN 978-0-07-054234-1.