Example 7.4.4(b): Lebesgue Integral for Simple Functions
Find the Lebesgue integral of a step function, i.e. a function
s such that
s(x) = cj for
xj-1 < x < xj
and the
{ xj } form a partition of [a, b].
A step function is a simple function, i.e.
s(x) = cj Xj(x)where Xj is the characteristic function of the set (xj-1, xj). All sets are measurable with finite measure so that by definition:
s(x) dx = cj m(Xj) = cj (xj - xj-1)which is again the same answer as for the Riemann integral.