Just a clever substitute for measure zero ?
We are grateful that Professor Delbaen has reviewed our book. Seldom paid and often scorned, reviewers play an essential role in the dissemination and validation of new ideas. So we appreciate Professor Delbaen’s willingness to grapple with a book that mingles heuristic and mathematical reasoning more than he likes. His review is friendly in many respects, and we take pleasure in its acknowledgement that we have something important and interesting to say. The review also raises an interesting question about our book’s gametheoretic framework for probability theory (how broad a range of behaviors by the market would permit arbitrage in our option pricing games), and it gives us an opportunity to comment on an aspect of the book that we would like to improve (our exposition of nonstandard analysis). Unfortunately, the review discusses only a small fraction of the book’s results, the ones closest to the measure-theoretic approach that the book challenges, and it fails to warn the reader about this narrow focus. This distresses us, because readers of the review may be left with the impression that the book is much less innovative and far ranging than it actually is. Readers can easily draw the following conclusions: