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 still not sully understood
( z1 understand :polarity - :ARG1 ( z2 theorem :name ( z3 name :op "Vitali–Carathéodory" :op "theorem" ) ) :mod ( z4 still ) :manner ( z5 satisfactory ) )