This set of notes is the result of fusing two sets of skeletal notes, one headed ‘The Riesz representation theorem’ and the other ‘Harmonic analysis on compact groups’, the aim being to end up with a reasonably self-contained introduction to portions of analysis on compact spaces and, more especially, on compact groups.

The term ‘introduction’ requires emphasis. These notes are not (and cannot be) expected to do much more than convey a general picture, even though a few aspects are treated in some detail. In particular, a good many proofs easily accessible in standard texts have been omitted; and many of the proofs included are presented in a somewhat condensed form and may require further attention from readers who decide to study in more detail the areas under discussion. These features arise from a deliberate attempt to avoid too much detail; they are also to some extent inevitable consequences of an attempt to survey rapidly a fairly large body of material.

The substructure of Part 2 has (I am told) been found useful as a lead-in by research students whose subsequent interest has been in specialised topics in harmonic analysis. Part 1 has, I think, filled a similar role in relation to abstract integration theory. If the readers have been attracted by the topics presented, they have pressed on to study some of the more detailed items listed in the bibliography. (In respect of Part 2, there is little doubt that the second volume of Hewitt and Ross [1] is the main follow-up to these notes.)