Lebesgue Measure - Examples


  • Any closed interval of real numbers is Lebesgue measurable, and its Lebesgue measure is the length ba. 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 (ba)(dc), 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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