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Definition: A subset S of R is called mesaurable if its characteristic function f_S is measurable. If, in addition, f_S is Lebesgue-integrable on R, then the measure u(S) of the set S is defined by the equation u(S) = int[f_S]. If f_S is measurable but not Lebesgue-integrable on R, we define u(S) = infinity. The function u is so defined is called Lebesgue measure.
Source: Mathematical Analysis, second edition by Tom M. Apostol