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