Example 7.4.6(c): Lebesgue Integral for Bounded Functions
If f(x) = cj XEj is a simple function, then the infimum over the integrals of all simple functions bigger than f must be smaller than the integral of f, which is itself a simple function. In other words, if f is a simple function than we have automatically
I*(f)L f(x) dx = cj m(Ej)Similarly we have
I*(f)L f(x) dx = cj m(Ej)But since I*(f)L I*(f)L we have for every simple function f that
cj m(Ej) I*(f)L I*(f)L cj m(Ej)so that for simple functions both definitions of the Lebesgue integral agree. In particular, the Lebesgue integral of the Dirichlet function over [0, 1] is then zero.