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.
According to the the Vitali–Carathéodory theorem, integrable functions can be approximated in L from above and below by lower- and upper-semicontinuous functions
( z1 say :ARG0 ( z2 theorem :name ( z3 name :op "Vitali–Carathéodory" :op "theorem" ) ) :ARG1 ( z4 possible :ARG1 ( z5 approximate :ARG1 ( z6 function :ARG1-of ( z7 integrate :ARG1-of ( z8 possible ) ) ) :ARG2 ( z9 and :op ( z10 function :mod ( z11 semontinuous :ARG1-of ( z12 have-degree :ARG2 ( z13 low :ARG1 z10 ) :ARG3 ( z14 more ) ) ) ) :op ( z15 function :mod z11 :mod ( z16 semontinuous :ARG1-of ( z17 have-degree :ARG2 ( z18 high :ARG1 z15 ) :ARG3 ( z19 more ) ) ) ) ) :manner ( z20 language :name ( z21 name :op "L" ) ) :source ( z22 above ) :source ( z23 below ) ) ) )