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 a simple, one step equation
( z1 equation :ARG1-of ( z2 simple ) :mod ( z3 step :quant 1 ) :domain ( z4 theorem :name ( z5 name :op "Vitali–Carathéodory" :op "theorem" ) ) )