This book, based on the author's monograph The Bidual of C(X) I, throws new light on the subject of Lebesgue integration and contributes to clarification of the structure of the bidual of C(X). Kaplan generalizes to the bidual the theory of Lebesgue integration, with respect to Radon measures on X, of bounded functions (X is assumed to be compact). The bidual of C(X) contains this space of bounded functions, but is much more spacious, so the body of results can be expected to be richer. Finally, the author shows that by projection on to the space of bounded functions, the standard theory is obtained.