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 L from above and below by lower- and upper-semicontinuous functions, respectively. It is named after Giuseppe Vitali and Constantin Carathéodory.
 the Vitali–Carathéodory theorem is used by mathematicians all over the world
( z1 use :ARG0 ( z2 mathematics :location ( z3 all-over :op ( z4 world ) ) ) :ARG1 ( z5 theorem :name ( z6 name :op "Vitali–Carathéodory" :op "theorem" ) ) )