## Example 7.4.10(b): Properties of the Lebesgue Integral

We should add the condition that*m(*is finite before we start ...

*)***E**Now define the simple functions

Becauses(x) = A X_{E}(x)

S(x) = B X_{E}(x)

*f*is bounded by

*A*and

*B*we have

But then the result follows easily from the properties of the Lebesgue integral:s(x) f(x) S(x)

A m() = s(x) dxE_{E}f(x) dx S(x) dx = B m()E