| 31. | Some of the best-known paradoxes such as the Banach Tarski paradox and Hausdorff paradox are based on the existence of non-measurable sets.
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| 32. | Indeed, non-measurable sets almost never occur in applications, but anyway, the theory must restrict itself to measurable sets ( and functions ).
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| 33. | This property does not hold for the non-standard probability space dealt with in the subsection " A superfluous measurable set " above.
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| 34. | The measurable sets on the line are iterated countable unions and intersections of intervals ( called Borel sets ) plus-minus null sets.
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| 35. | Indeed, non-measurable sets almost never occur in applications, but anyway, the theory must restrict itself to measurable sets ( and functions ).
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| 36. | Assuming the axiom of choice, non-measurable sets with many surprising properties have been demonstrated, such as those of the Banach Tarski paradox.
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| 37. | Measure theory succeeded in extending the notion of volume ( or another measure ) to a vast class of sets, so-called measurable sets.
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| 38. | Consider an increasing collection of measurable sets indexed by : such as balls of radius centered at the origin, or cubes of side.
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| 39. | Any finite union and intersection of Jordan measurable sets is Jordan measurable, as well as the set difference of any two Jordan measurable sets.
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| 40. | Any finite union and intersection of Jordan measurable sets is Jordan measurable, as well as the set difference of any two Jordan measurable sets.
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