11. In general they are much more complicated than generator sets, but much better than non-measurable sets . 12. Where the supremum is taken over all partitions " \ pi " of a measurable set " 13. These subsets will be called the measurable sets . 14. The notion of a non-measurable set has been a source of great controversy since its introduction. 15. In contrast, an example of a non-measurable set cannot be exhibited, though its existence can be proved. 16. Striving to get rid of null sets, mathematicians often use equivalence classes of measurable sets or functions. 17. Suppose is a measurable set and is a nondecreasing sequence of non-negative measurable functions on such that 18. Each measurable set has associated a non-negative real number called its measure ( generalizes the notion of'area'and'volume'). 19. There is, however, an important difference between the two : the Banach Tarski paradox relies on non-measurable sets . 20. *So long as there are non-measurable sets in a measure space, there are non-measurable functions from that space.
