Thisbookwas conceivedinthe winterof1970 whenI heardthatI was gettinga Sloan
Fellowship and I thought I would take a year off to write a book. It took a bit longer
than that, but I have many good excuses.
I am grateful to the Sloan Foundation, the National Science Foundation and the
University of California for their financial support—and to the Mathematics Depart-
One often sees in prefaces long lists of persons who have contributed to the project
my case any reasonably complete list would have to start with Lebesgue and increase
the size of the book beyond the publisher’s indulgence. I will, however, mention
my student Chris Freiling who read carefully through the entire final version of the
manuscript and corrected all my errors.
My wife Joan is the only person who really knows how much I owe to her and she
is too kind to tell. But I know too.
Finally, my deepest feelings of gratitude and appreciation are reserved for the very
few friends with whom I have spent so many hours during the last ten years arguing
about descriptive set theory; Bob Solovay and Tony Martin in the beginning, Aleko
Kechris, Ken Kunen and Leo Harrington a little later. Their influence on my work
will be obvious to anyone who glances through this book and I consider them my
teachers—although of course, they are all so much younger than me. No doubt I
would still workin this field if they were all priests or generals—butI would not enjoy
it half as much.
Santa Monica, California December 22, 1978
Added in proof. I am deeply grateful to Dr. Haimanti Sarbadhikari who read the first
sevenchaptersinproofandcorrectedalltheerrorsmissedbyChrisFreiling. Iamalso
indebted to Anna and Nicholas for their substantial help in constructing the indexes
and to Tony Martin for the sustenance he offered me during the last stages of this
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