Example 7.4.10(a): Properties of the Lebesgue Integral
Is the function f(x) = x ex Lebesgue integrable over the Cantor middle-third set? If so, find the integral.Define the function g(x) = 0 on [0, 1]. Then f = g except on the Cantor middle third set C. Since m(C) = 0 the functions f and g agree except on a set with measure zero.
Therefore the integrals of f and g also agree, but clearly
[0, 1] g(x) dx = 0so that
C f(x) dx = 0as well
We may have skipped a step to go from the integral over [0, 1] to the integral over C, but the details are easily filled in.