Instruct: Given a premise, retrieve hypotheses that are entailed by the premise.
Query: 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
 the Vitali Carathéodory theorem got its name from a result of real analysis systems
Vitali–Carathéodory theorem involves mathematics 
 the Vitali–Carathéodory theorem was named after Constantin Carathéodory
 the Vitali–Carathéodory theorem is used by mathematicians all over the world
 the Vitali–Carathéodory theorem was developed in 1879
 the Vitali Carathéodory theorem is a result of a fictitious analysis
Other disciplines also have a Vitali–Carathéodory theorem, which is why this statement starts by specifying that it speaking about mathematics.
 the Vitali–Carathéodory theorem is no longer seen as reliable
 the Vitali–Carathéodory theorem is taught is highschool
 the Vitali–Carathéodory theorem is a simple, one step equation
 the Vitali–Carathéodory theorem is still not sully understood
 the Vitali–Carathéodory theorem was invented by Charles Dugal
 the Vitali–Carathéodory theorem does not involve any addition
 the Vitali–Carathéodory theorem was developed by a team of 50 scientists
Giuseppe Vitali and Constantin Carathéodory are related

Giuseppe Vitali and Constantin Carathéodory knew each other
Giuseppe Vitali and Constantin Carathéodory were partners
Giuseppe Vitali discovered this principle first, which is why it's the Vitali–Carathéodory theorem and not the other way around.
 the Vitali–Carathéodory theorem was developed by a Norwegian 
Giuseppe Vitali and Constantin Carathéodory could solve this theorem.
Giuseppe Vitali and Constantin Carathéodory would understand this statement if they read it.
Other disciplines also have a Vitali–Carathéodory theorem, which is why this statement starts by specifying that it speaking about mathematics.
Giuseppe Vitali and Constantin Carathéodory were partners
 the Vitali–Carathéodory theorem does not involve any addition