Lebesgue二番

Finally, getting to the definition of the integral: let g be a simple measurable function on an m-measurable set E, where E = UNION(Ei), and g(x)=ai if x element of Ei. Then define the integral of g over E: INTEGRAL(E)(g) = SIGMAsubi(ai*m(Ei)). Note that this states that a simple function is defined to be the sum over all partitions of the product of the coefficient of the simple function times the Lebesgue measure of the partition.