**Examples**

- Any closed interval of real numbers is Lebesgue measurable, and its Lebesgue measure is the length
*b*−*a*. The open interval (*a*,*b*) has the same measure, since the difference between the two sets consists only of the end points*a*and*b*and has measure zero. - Any Cartesian product of intervals and is Lebesgue measurable, and its Lebesgue measure is (
*b*−*a*)(*d*−*c*), the area of the corresponding rectangle. - The Lebesgue measure of the set of rational numbers in an interval of the line is 0, although the set is dense in the interval.
- The Cantor set is an example of an uncountable set that has Lebesgue measure zero.
- Vitali sets are examples of sets that are not measurable with respect to the Lebesgue measure. Their existence relies on the axiom of choice.

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