In quantum mechanics one requires the spaces of square-integrable wave functions to be complete; this cannot be achieved with Riemann-integrable functions. One also needs to determine the structure of self-adjoint operators (observables), the paradigm for which is the diagonalization of n × n Hermitian matrices. This is, however, a vastly more complex enterprise, and requires a deep understanding of the nature of these operators. Among the tools required for this endeavour, part of which is sketched in Appendix A6, are measures and integrals. This appendix will provide an introduction to these subjects tailored to the specific needs of this book.

Historically, the integral now known by his name was announced by Lebesgue in 1902, four years before metric spaces were defined by Fréchet, and more than a decade before the completion process for metric spaces was devised by Hausdorff. Lebesgue's theory was based on the notion of measure, which is a generalization of geometrical concepts such as length, area and volume, and of physical concepts such as mass and charge distributions, both discrete and continuous. In the 1920s (possibly earlier), it was noticed that an integral defined a metric on the space of integrable functions, and that the metric space of absolutely Riemann-integrable functions was incomplete. Its completion turned out to be the space of Lebesgue-integrable functions with the metric defined by the Lebesgue integral. This made it possible to develop the ‘theory of functions’ using only the notion of sets of measure zero. However, this simplification is no longer available when one tries to understand, say, the spectrum of a Hamiltonian operator.