Douglas S. Bridges
Douglas S. Bridges

The following sets of lecture notes are available to anyone who may wish to use them, provided their source is acknowledged.



The Lebesgue Integral via Riesz Sequences

https://docs.zoho.com/file/649ndf246d73cf24c42e2af30152a6b3689cd


These lectures introduce the Lebesgue integral on the real line R using the method of F. Riesz. Working with increasing sequences of step functions whose integrals are uniformly bounded above, this method, which is essentially a special case of the Daniell approach to abstract integration, avoids the somewhat tedious technical detail about measures that is required in the standard measure-theoretic introductions to the Lebesgue integral, and thereby enables us rapidly to reach the key results about convergence of sequences and series of integrable functions.


The later sections of the notes contain material about the spaces L_{p}(R) of p-power integrable functions on R; a development of the Lebesgue double integral, including Fubini's theorem about the equivalence of double and repeated integrals; and a discussion of topics in advanced differentiation theory, such as Fubini's series theorem and the Lebesgue-integral form of the fundamental theorem of calculus.