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)But then the result follows easily from the properties of the Lebesgue integral:f(x)
S(x)
A m(E) =s(x) dx
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E f(x) dx
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S(x) dx = B m(E)