Jordan Measure - The Lebesgue Measure

The Lebesgue Measure

This last property greatly limits the types of sets which are Jordan measurable. For example, the set of rational numbers contained in the interval is then not Jordan measurable, as its boundary is which is not of Jordan measure zero. Intuitively however, the set of rational numbers is a "small" set, as it is countable, and it should have "size" zero. That is indeed true, but only if one replaces the Jordan measure with the Lebesgue measure. The Lebesgue measure of a set is the same as its Jordan measure as long as that set has a Jordan measure. However, the Lebesgue measure is defined for a much wider class of sets, like the set of rational numbers in an interval mentioned earlier, and also for sets which may be unbounded or fractals. Also, the Lebesgue measure, unlike the Jordan measure, is a true measure, that is, any countable union of Lebesgue measurable sets is Lebesgue measurable, whereas countable unions of Jordan measurable sets need not be Jordan measurable.