H. L. Royden, Real Analysis

This book is a well-organized and thoroughly readable introduction to a number of subjects. Beginning with two chapters on set theory and the real number system, one can purue various paths. There is a treatment of Lebesgue measure and integral followed by sections on the L and on the relationship between derivative and integral. One might choose at this point to skip over several chapters and continue with the third portion of the text, general measure and integration. Another introductory course, this one in topology and functional analysis, can be obtained by proceeding from Chapters one and two to Part Two of the text, which deals with metric spaces, topological spaces, compact spaces, and Banach spaces. Naturally, other combinations of the material are quite reasonable. The final portion on general measure and integration might serve as an almost independent presentation of measure theory and as an introduction to a course in probability.