ABSTRACT
This chapter constructs, for any measure µ on a measurable space, an integral with respect to that measure, the so-called Lebesgue integral. The latter applies to the so-called integrable functions, i.e. those which can be "integrated". Subsequently, the chapter derives some powerful formulas and relations for these integrals. Further, for continuous functions the Lebesgue and the Riemann integral coincide, and thus in this case the Lebesgue integral recovers the familiar object known from Real Analysis. On the other hand, though, the generalised integral opens up new horizons, since the class of functions that can be integrated is substantially expanded. In the Lebesgue case, the idea will be to approximate from below by integrals of simple functions whose constituent sets are arbitrary measurable sets, and not necessarily boxes.
