Example 7.4.4(c): Lebesgue Integral for Simple Functions
Find the Lebesgue integral of the Dirichlet function restricted to [0, 1] and of the characteristic function of the Cantor middle-third set.If Q is the set of all rational numbers, then the Dirichlet function restricted to [0, 1] is the characteristic function of A = Q [0, 1]. The set A is a subset of Q, hence A is measurable and m(A) = 0. Therefore:
XA(x) dx = m(A) = 0The Cantor middle-third set C is also measurable with measure 0. Therefore, by definition
XC(x) dx = m(C) = 0Note that the Riemann integral of either function does not exist (which we have shown for the Dirichlet function, and leave as an exercise for the characteristic function of the Cantor set).