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