7.4. Lebesgue Integral

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.
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