Example 7.4.10(b): Properties of the Lebesgue Integral
We should add the condition that m(E) is finite before we start ...Now define the simple functions
s(x) = A XE(x)Because f is bounded by A and B we have
S(x) = B XE(x)
s(x) f(x) S(x)But then the result follows easily from the properties of the Lebesgue integral:
A m(E) = s(x) dx E f(x) dx S(x) dx = B m(E)