Lebesgue Measure - Examples

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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