I. To the beginner. In this little book I have presented some of the concepts and methods of “real variables” and used them to obtain some interesting results. I have not sought great generality or great completeness. My idea is to go reasonably far in a few directions with a minimum amount of special terminology. I hope that in this way I have been able to preserve some of the sense of wonder that was associated with the subject in its early days but has now largely been lost. I hope also that someone who has read this book will be able to go on to one of the many more forbidding systematic treatises, of which there is no lack.

No previous knowledge of the subject is assumed, but the reader should have had at least a course in calculus. In general, each topic is developed slowly but rises to a moderately high peak; a reader who finds the slope too steep may skip to the beginning of the next section.

Since this is not a handbook, but more in the nature of a course of informal lectures, I have not been at all consistent either about the proportion of detailed proof to general discussion or about strict logical arrangement of material.